%!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 28 /lessmuch put dup 29 /greatermuch put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 59 /emptyset put dup 63 /perpendicular put dup 65 /A put dup 68 /D put dup 70 /F put dup 72 /H put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.00 %%CreationDate: 1992 Oct 23 20:19:17 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def /UniqueID 5011949 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192e3f6620e5383410 9c674a944e5ff85290c767088ede1e6835bf2aaf7a0e947193d98442808fc3dd 148069ab44a71076493a05cbf836aeb3dd5c78829d1482b3c40c2f4cf5fdd0be c3c3c1e08cd5d21db9363551e5ea23a95a928a532e09d2b59879067059a5974e 4912c690ee71858bbf009a75913237a9ab9eb9d8d683ea3fe066fb2ecd3df7b5 c7cb950bdefac7931a3501dd1dd59c10d45f8f91ff07c7c41361e1fafb605c38 0431bdf3b1e44d21d351cf80d8dc6fbb5131fb8ad1a0e9a9a5fdea6432ec1d59 bee32029965fb5c141768d6f2c624b2dae5c9ede58cb4f522099a96749094f15 07e9a6b977ebc86c877b8c4a0758b69dca966be63e606c8efde1f65b295997e5 1205bef471d087f51dc9ff816ef9f6a4a23226f9305feba8da8baa909e515d85 b60901cd8e958e0c94d2a64c1d9510e66d37d18fdf08c5408032c63ae7a667bd 1ca377a9eab6b0bfd434ccc71b779a6cbab2183016bddbc05f171719794bf19d 0118b9e708296d876e7ebd011b0d49e51b56e10bae4b5f2d07b16e0c83df55ab 42cf85da87350d752cd4522682b02f82c49d08a2e0d47d58374443a63831e432 c48f945cc5e42d807ba86cf3e7b3b09f636c17360c5c9bd7717f7e4909e8bb4a 7efb7a580cc9357613467b64fd0161dc026602cc382bdc1779ce8ebc0587f172 654aff1ed908166f3755ccdc64a183e50dc8ff8b9d8ea9ab556615ce390b4660 1749955a3ac5f7c061d6b9948677ca68373ee7c6738abdd0856c90a91d904c91 914f3ba09a3a8b54fa319b1ba9bbd7441b69f0431716a4156816329b7e09b57f 495591d0d9c9049cd0475c7d2b75a8105b99f34816c75f82922b2ff19f835843 adf237dbe7d8fe6c10f1a7886b55baabf55cb33784653da0ac3b49692b348e26 0e2995ae562bf8fce2a4754886dfaac090986bb50eeff294d171eff1b12356d2 e67a7040f2ac7eaa0fde3cce8808b041589918d213923d556f42dbf5c27a062b 4885c544d36fb68419aa660bc55824a647ee56d3a4f4c48ee9b6168c721b54fc e8c202833282c38447a03eebad4a0098fabff640b27298f9e63dbca777a37849 d70d0ae5436463a8d8272bfbab7dee85243e4046f8af3fb02dfbf7fc9fd59837 9b3af8cb2faf3a58d7aed120e3c05f5fc69074b299057f821eab 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 82 /R put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 1f72a7ea6a698d9580bcafcc2366c5964e5d3094b67d86eec82e16761e4b6f4d fd245d3c0b4188a0543ee88fa787fb10793268e415e389ff071cdd46bf6b260e 1e9c9ac037edba327c6379836655308ddf35e4a4488ad32d3ab301c67a2fa71e 8ae00daeead4271df276f5ce395a57e414f85237cf8f1be070309ed5c3078506 c9d38c2adeeac618d04c0be2ff891dcfc9ada7588efb91f739188a933e1d37bf 632cf042106df1bae0cdfb79c578e565014cead123bf856cc0c14983f5548a20 c0d45446e0dd2b017f528e46d394a88a2d5f86fc669ac7d489f1851c100b29a0 ee3ec82faf87eba944a57dc14ebc94f2445a8a21ee68eef1848d534bdd192ea1 181220351ff93c51e53fcd5ee6ec92bd644d121eac52bc163acb6948a7a4c256 7243c7113b1b5650aeeedb14ffd9b6ce33605907595230a06db01bfb510a460e 76598fb2bb67498f79f0f5b5c5fdb5515cf2257aad324ef1a877a296e2851d38 f8995123e527c4956247fc8ab910206a9ce8736563ff0d4cd5d00579d60bca51 650af348ac4d6422c4cd74d924b532d4a28ba95eff24aca4b3cd1d28e207c7f8 906f8dd99aa04c135fe46c1dc8ee89bd768df17240a70f1a96382defcc180b9a f1b6fa57a9b0a2440bae0797790f91733970b16e742e6341d95faa40c583dac5 13a84eb181d2946b5d0075f71db8a991dc1a809cd3cfec573514392567a82e75 946ccf6e14f4a968313b3b6d727c83018ed58b882b96db208ebd27f7d30ffbc8 ab11e10a0f81425a62bdf7eed9f618c746103dc2d1c3e1f37d27083f4068a30e 099c92685e320266ff8047e81140658a63a6b52e4cd63e9403c28a892c8d258e 39492535973e1c042174e433ef44ac9cc424e2573468eb807994587b8f94c4a9 b735ef79826c5a339bc35de1b5798ad05ede0b5f6ddf75fd73fb696e33d94ed0 47745e811685ba44490207e0234e657d1dd4796177f7c097fff02b080fb3d738 76b62656162b436e61c4672a0e4ba26437858e470efdc12ff2864a2e731e2325 641c23740f725e58373eeedf53f1c9cfc285add81d2d5e3dc2a1297c3135d5a8 199c5c34f91f9f8344dbbf4ca3772abfdcbf79145a0b95ac128a8665f42c876c 17aaffede87efa716562b9c25bf63056c9739fd972f76af580bd0ed5d5e11c65 033e553128e19f3b57a0c02a1a65134310419063e9ed666d0c3d5d0bee01bb7e d574d828145b2ffa2cf54346d80a74ef004365638625f1df7ea88d2fda9bf39e 172288f4aeee9e9d4fa041c7ebd1c9b047d8039c456bd03916701d320ecb4c17 1becfef5a82a501f35fecc33c1832f3abe40f32560849f3190ffb17b25079ce3 85758693b385d2ac0b24c72138b316311479cae513b2176f27905b3842a9b947 c7c8bf7740ce40f1eee95c65cbcf241e6bc976244392d9ac0947d098ca0f0ab8 d2ccc9b93cf74d4452954f1929e6266cbdcc0c9f00bf7f5ee7eaed83ca6eac74 644980ec005b5bc680e84adc95083b25ed5e5e8fc2cc8cf9ad4d07947e2e3b64 afbb829574502f7e4549a47de262bb3d968308413058836d1e33d170a15ea7a1 c85607d5d208056750d95a769fb3ebb150cb259410a8be76a4d045dddee690cd c01d6fab4abb0f28a2b5b63e599c6eae0b80efd639fe92223514b9bf3c3ba505 cf4fd92b17f45c078b5cc1a01aa4704b918bd3afc9a52f45ef989909eb2a8a4b 3cbddf7be5ddf7f941fdcf510103da02028f1729fb6bd1ce088efb91bf709b56 176f965e2576ca2177d3da01a5dd47437fdea4fb5664b93093079ce2698faadf 2818e551910ed17f812c9cec0fd3c8c31602ee41d87b76b4fae73e23b0f35ef1 fe5f2cc10abe11d38c2f0098782077c39aa41eeec04d9d2bc3ebe67a2ba2e8ec 24a58a4865002c7af9733f3b77ae7b8fa9e6d1ccab2ec4e5dfd9c8dfcc1b3138 615cbdf2ecee900c659aad98cead5a7abb18fab42a26e1a320902bc6c298e3a6 fa096d9c1b4fcc6596068fa4fbddaebe22f03f268759ffe1ac283d6f1750aee5 760906c98d6a0540b6da2937e08d7014089d4090eaa741f5f4fbf7eb13f844ea 6811c6247faaca51f5060817348b03ffd305465fef78dd1bc0615cb9dda52ffb 97f01d14cab61d1323cb82493b724e3580733299df6f0efbca2e5d90e58877e8 15ca2cabe94fe4ddaaa7694295aab2e47fe02a8c7d0e68f4bea1cf8c2f1b574f a2fe5c2556075dfcb2b3d5e9296fd21f2a6b9743e7b11cc558f6fb71971f97c2 55b37fa1aacfe52d2aac9f5c3a30b7d131722495d032ca4fbd84ba8a8988ebea 4a87358199da911697ab2c727e2d0df0d99776f64d06d3cf1578a7e08065bff6 2bb173e1556e8f3ceb3d4e25c15745be0cb0929d79b0ea646f1d5def863e1efa 2225951cf487593ad6ecabd12534973ca588f038333851514c6e9c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 100 /d put dup 108 /l put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 9390b8b2d3a0204a6fec9e5a25a634ca16bc351f1189ca97d5cd1af3e38d04bd 4e5cbb6ff6f86576017b16fec97ab37dfd6aa6df3b77dd4343fc9550a4108b62 b91277f3bac333db6e2618320325c0b8e92cf0c72caee4bbeb0f1b3ae212ed86 1b07a8085d47219056190b26d12b4543e77ee939670a64e0e13c414ae6ac5b9d 81dff84326197e50ff251c5573b9b7962028c77cfe1f29f288f363ad6d50a58d 41543a32bd80d8f80b7b75530204400257c2c9c4c68d5736506f855bd55d7b8f 2ddb613cb63022ccb0fa36cd38272473f96b8245d010b531322e119b1b78be8c f6d7f4eabb1d4df86dc3a0e5eae61c8b80f59bc8a4fcfd09a8d8066cffa7bf5c 2ea014015bf4436c7f193bfbf876e42687180d4a839f2fbc4998e58e1eac3c47 7e69e93c44dc5e58f7c0753d967a89c29d223e71fc2be59eeae47c50865ef6e5 1c7cd3770a91f3580bcd9f70e5842e712351e59193f8d8306bd4958903b9b75d aea6e23afe805f57930b9fbe5ecfd3b3d7dd5d16d3da0605574cb7ccc2c98587 9f9b29629c78793bb8670c32b244534f42b16ddcaf5396b649dd482b4398fb6b 72b12525be8d6e17439800ec61cbbedf065d77e20f101025da07d986bb288141 9071d9529aa403d3d37e683bf7cef17f5bd230ddc0f25dd754d74989839697bf c8261aad0e2609988d0bb7bcf82c11f482dd5246454cb78d4c81a01beb67a2f5 dfe954cbbe4a78a88726dabccc16 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: msbm8 %!FontType1-1.0: msbm8 1.2/19-Jan-95 %%VMusage: 32482 33913 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.2/19-Jan-95) readonly def /Notice (Copyright \(C\) 1994, 1995, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-C.) readonly def /FullName (msbm8) readonly def /FamilyName (msbm8) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /Descender -194 def /Ascender 694 def /CapHeight 683 def /XHeight 431 def end readonly def /FontName /msbm8 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 82 /R put dup 84 /T put dup 90 /Z put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-59 -474 2478 974] readonly def /UniqueID 4702307 def /StrokeWidth 0 def currentdict end currentfile eexec 9b9c0887db83fb1ecd8335b0bb39cef0af64f656fc6e5c230cc9d3a7346aac7a 06bd9a40393ca15d3773a21e06b9b4254d3050e90726bbb09120935a8d448cdb c799d90205a21291254fa633cc0c2ed88781ef21a5d45b72445c284fa44f8f39 cf566976075a92e9e2947a6fc93d879c29eb26809ecb4409cbc8666526cced92 65de661a2f8b0a16be45a9da17eeaf89b0e023de2b0373dcab997d60b7d7dc1b 4f7e650a2a9f13ca0543f3b080ae33d65f2802b8c6f032ed6f1f9ec1620e5d5e ac22d6b7a2c527d76fc035bf25a0299fd60a72ae7e0c045feb341646af23c463 ed2a5dc1919c6ff451154c679acff08a89ce1df2e249199fa1bda096e03fdf15 4361c76bdfa752edbddd62db3ab84a90e2bb9721b75b59a62e3ca79118748de0 4e0e6e26634604d253f71331cc6ba447a8a447031b1f313982407233d775612a 4a7724697c7493bcbf0472b29db0afd695aa19d59e0dac4be39cbf9c23c739cc cec6365429ea94ee6f803a5344c5f3f7cf9df9b4beedf4fbf49bebce757e1d8c 2a21c63d62f82f5dda542d838c67ec6f1bf39ae00a6d30401992944170e200db 493a28d6eede182c3d037a16efe0d9181729b170ccb04ec3ade7dabdc6b43e82 608979359b287adcb6d52fbdd08fc869310ca7729a48ad2b7624c662a80c6660 2f384ab69eaeae82818175fdf810bf18936efaaf360a316bb572f2ac44a3b67e a75d6dcc5a7df3de860cde7ef589080fae1f3203406b3feb87f4a8dc03273467 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18bcef8ce86cf41068e57e53e93f257fc94189ada8ff92066c2c07fd9fc52582 4111ac513dc24b7f64d243fb6f7d51653b76072e645a135bd13ecb6a10d6f172 c071c2e2b7b5efb3e432bfdfc9bf80e9ff01b22c163f001efb760affcf46c617 e06118544283541cc356cae83239994bd4238153a7f2c2b862f6c082ef331387 78570ce37f7b8464b6eb9470741ebfbfbc 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 19 /acute put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 53 /five put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 127 /dieresis put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 3e224d18baa726369542682e9459b951b2ac638494880c42be4e2175a3c067e5 945bfc1a05d3de8c3e8ab97f80ab6000e1f78fc258b85d3c84b038d7d0f5be85 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%%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 73 /I put dup 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c a595df78200c352997ec3ee564df229fbb5473f5e8ccb1cc0153e9a7e299a8ea a29b69d1b622b1f0cffc58291248759607d91150cb0651120970dc9f743bebef 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 99 /c put dup 115 /s put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523da5d7776ec0bb452 fc23761a3e881b097d783e66178ac7cf9fa738d9d6b9e2765f7320d6fe1d56dd dcdd484c4ad023d1e878edfcbec88cd8d25038bb6003256519572eb20468ef5c 62f33d8707ddf81af347ee2541ae92a0b38f0c8cd3c4be97c1717a07f4e6f0d0 3d9b643a12afb192f70a3e083508fb65c17a9d10912e07b555fa53127e5ddfad 4257dbf90efa0e0f8adaeb5b1b818782c2446dce70bc7233881ec1b358418d8b 18992c9b49f5f05a5a05361cb800cace4105c7920f437e4b765f8c83b50b0225 113b8a7d5dcfb8227abe989d62a0a47244ffb1ed905345634653192fda9ac11d 0d8368be52afa43b84202033d2eb164bab7cba9790bcda96838caf6bc43e4b8a 04c911d81cac5e21dce6e497bd61e33dedf8d1acea5b117b54c9d9177a4465f9 9e2ae71bdf71669910459d2383959877382a087eb701b861c59e4a33bb46f464 e856c09f5ba81e58561c2d232171905c0f5c9f27f7618ccaa840aefc08dddb1c a5f8602b65a9e4d14f4c1aa3e1c82c75a823bd73b2fb8297c7de88ed62cda9e0 cc91e1a4e796c982236265c4eacf8e26f8818912f2011858fd677b4c00a9788b 3bbd855ddae79238d002e57ca43a02ca8c7527343cfdc7516f0845a04f231ebe a7c3ce3bd683a40c4f48dc2743c7fadb59fab89d04f77a84cc270dd7df84bd28 ba3f37247d9a1458b6a06b63aecfd50138f389e84af6a75a6a9c677289f9ca8a a051e9caf30b9aad9c3a4b22f0d0a52f9468769c268c2cf366bb65919a4b2805 67c87ef4e7edf19d69f51f51f56cd558c8cac4e543af99dd560c78043c3580ed 6f49dd712b54a5f82d8b700c19445e374a2bb1480dc87e9497ee6e5027639d9b b0f9e07716c8f3de4724cf1b5148c85d26f1c3ad265a180f0027d24e3a00b48e b0cdd9f8703afc39c52a6895e36c2e5de9abcda7a682492628c1b099c20f0efc 8bc99c28ca7083ee2847f006ffaa7322e0da5589892d1e7f0d50ddd16d6f747a 11cc010eec8aa51c1011fd2fa50b7a6290ac7c89a6be361b05178d3f38b39b86 c3410165963c80a0db07aa99b2d082d413f3ae2ac249678a646a63ecee76317c dc65d710eadbc4ee34821979ec9d759fc71aba79f0d4c18089cde263a04925f7 2a8eb79182b60e4fbd2aa4c285ba1d3bed4cc8a8947501387a022e599c4a6d23 93bb402d373bcaef9fefac127bf3f99503534176413f42e3d093e3017095eac0 2c208c7ec5c78437530c62e0ea42f1d62c7a48905f6c107ab7e3a91477d239c8 53136be3490fd1f5836be073d4bddb54fc25d30184f4983aa124696ae7513dc3 286d1489adfb0efe619317d9b3df7a4aa536d99ade0a5bf9518faeae1f0da6d7 2e763a38aa2a8bb427c5c48959e6630118a24060e8cf9b12657c650f859ebb42 52c8b8f699a02a716035e1ed7653bd737500 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 6 /plusminus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851905abd26842613d8fb644512275d43f312eea72f7ea3ba90c8fda6ce 7a9dacf52f55f0dd557f8d866ed6f3d3d5e6fe7e9ee79404fbe1a285bc519ecd 7ca162f0729044aff526bfe554ed59dd9beab0f842d55be08caada6136ac9f94 4cd96ff1c2f4927d87fea4496a1d612d58eba570eef0007c665fd143e9030923 517ac31aaf6912012671dd87686f2aaab25a2adc7501663d8a1ef80cc7da7dbe 13fb5333a4618ad8b4a6c2cc8fa3d616c65848c95ac10893c62198f642f2d2f3 503204e5d5aaef2602d4953fd4b77b402232f603d83de73bb3c2d50e2416481d 92e8147eef905c2812d6f99a347d02a88fab1c99dfc7a239e35262b5a40dbdef a0811fec7b3bd9cf2fcafcc714d3a5af1169f908d3df07e8705961d038e1881a 4f69eb69c9df1c78540f42b37de0ede2647d115acac4dd8dd54302911ed1dfd8 1ade02edfee46f8e3c5de5159938e4f5076a66243838a28e42b5f757e5fa2223 c5b56414540510a396366fe857023eeefa3b4c970609ea0ee08ba4dc6a1e57fd 6f3a1ee5257880b5d85e32bad58be98f232d603ec894e282326baec250e7e9fd acc49ef9b1dc997b3fbdc6ff862e02b1a2baf7c00b39aa7d82002d7256d9ae 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 6 /plusminus put dup 7 /minusplus put dup 20 /lessequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 63 /perpendicular put dup 72 /H put dup 83 /S put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a42f5d4df9d7a11b6fe6b548448f164cdd108b0ad1b094bf60aceb79 fca8deb47b091c9508b14dd9ef4a98e676ea2d2f04cdbfba0fdbf53c0f62d022 91e3f977ae0917075c1ee033cbeb20e478b8e24c51c0fbcb65f26f0c4363e010 47e99cecc2d6422358d6e83b79a8c92551b55fc8cf52a5c305ad9e36814b37e1 dcd656d33df256fde32d33219b032b8cc03d097cf3d82b1282d556e08947c0b9 54f37de1396efbe4916739ebcb9593dcc386db6c8a3cf7e557d13976ecfe6061 48bfab61700de0aa995108bde8d5e2fc9c8358dc67c2a6cdb6a9ba8cff447f50 477d850bd93b245077387e3bf3baef078d85d28aa14574f4b3d070d319156c60 9aeb4b7655d09d536514eff2d2ea58fe1f1f2805fce5f13b34d8bea2ead58d2c cbd6bda4700e75a3d2d756721e5cd9a657451e3eaf037f66c6ee71ca68bd907b 2a873b5e846a6754f96968d11f9ace637eaccfb07a8291c560d60de23f483475 803b60777b7974c10f670e44a7378dd8dc7139ccb23b2dfcfff8a321ce2cec43 2145f514641a8c13fb009727bd30a051b845a3971ee4be8e6ee449fb10012b8a eb829b4f1c14e9cd254d9ffabe6df769c780e44c26d913b9e6137205d3256f33 7a1a1a656324cb0ec8d191e6dfc4090c12b8c6820f411e7906b95ea3bff3e453 d1f09dda9a2505a22158be321870d58d91b49b43af4def7c74e3fbba0e74453c 193c116fe46f3ac2d0c24ed8306d878f43ee1b4b226fe1bf32d99982c5b5ecad 55602b3336fbcc88ea49996742d17073c941a7adbd77724ece4556197fa0b903 5b53a8fbc67e743002c308d2d1b42853e041e81ee1b09b6fa8d17ece06e033a7 e357c4434b12493829a9cfd7b7104b085d349239d048014f7163056fea289a9b 5978ee4e0764bfb5c477a1b0fd6a008b99e71b6fd2d391e365432a65e795db22 f8925736d291a0598a61cbdbed1134012aa9353d54e201ea25aca303bebb260e 3aa1e063a24aed88bf43c4b2d8211c98cb0d69f504997a29ca44cccb62c5ac1d 2308c261dc65e56160bd09935581b1e584e146d652c1e3719114ad471fba4f28 2acc1048c8c935a41fbef78404ed7aa2e457044cb4902140af86f3186d85e627 9d53320ac9eb3ce6a47b5065c4981fe73084a690da20c91a790c4fa9f724bb34 d5e09644fd2bd06307da1b7a026fd9a3509eca160cce1c66d5ed8bf9d1fa81b9 cb74676bda8933aaf611a4d3001ff579f2c1e1906d51d5dcd1f2792457a2b9a9 c76bab2a80ebc4ff5578990c986b6d981df7aaa2b229bd912a06387b31c2fcaf d5316ae202fcec8ae5c9c85b6b73769f1813f7a816fdacb06b5315f40ae7cb12 047aa49592b20e6e20b4102a716d3ce00250e3d390e2964ba8437205bd02ceb1 6c164e90f61fd90889147a8beda04a62dec6ef98f64a65a9b8018777547df928 64f3305db7b119823868a11fd90da4a8868167351dabb735042ceff858957a6d 1b9394eabbde18a0b822f135ff5a78621cd42d307243337bb7d9f3071e5faa88 e931241aa1b79898a40a8473a58454101e0eca4f176e66c2e1ae922d4f82fc5c 1d325b4f3b582022f261de1cbe2e990c862027ac901211ca35dbc3212fcb0eaf aa11f23734871a3ce7ff2b891cc00d1c3cf232d1ad7920ff34aed24607769ec5 71f56dbfc2f229cec091c1f11d85f78bef397180602052c4b756e7596c36d514 7719e7346e5e284e25098eac5779468748fd697f58cd162b738426d0f1295a64 b4c1da96598de32f0bd54a5f543969bdb422596cc993aba936c8999ddec883b4 879bbdc387712390d088d2a6f1485705c73503d4856700923e31d4b7ef0cbf95 83891f5369737f965bf3df47a353e6199dd00528f84d68045c0ba6378266c745 035c061f26b0f478143e4b2badf38fb3df6a1f6655793f67882efedda292d8f8 513adf69c114e7c1c024c8eede3ad0d295465c8610e55f08406fe2e0fafa6fcd 5b28971d5697b38e4746bdd5f8840e543194938c0411fb9a6fb5eefacb560a6c 3764ff6f8e0a83cff141e2044ac3d6c797a233c19057148a3fd7efbba5fe555c 3a02fe968e01064cabb2a9fc2034079978c2464613aa354a383a0d0d51fe61f5 5452b171bf60f179283dddc06fe439a172b6088fde75be9e6e98dc713eaae58d de808ac9d1fdad14df601ab7cb93051d662f6536b7e2acf065248cbb1b62ec08 4bf5d6d9655615fc7eba5074d757ee7d95803dac387b59abf4449a90501404a7 da4d07e7630aabcf76d35753648556dcaad483c60cae4aa5b35d8554fc0bd6cd a9b9dc7f4dfcda1a26d252619443fc80076662eb5a0554e8abec0d07fa04119e bbac4769ca7cabe451f36cb54c350c0f91ef469a6287d8e3b8333e9e7b41ad44 53c4add2b93a999030b2d9aae04b2b58e10ecd7a939b2b978ea5f2f0ec2c3513 483e4d11e770278743fc7e8745dc21cf0b2338743e127b5e03f903cee41830a8 81bac5c3d1147962ebc6993c5ac927f0382625f42ca83dd6e67e863646ba76fb 68ce7b3bcaa9f29d7b4d53333399c11de8cd0653724f2f6ee62597489c15fd68 18ad8daa2eb360aa31bd25333d3049f6ef2a1cfd78a032e4d58e79b0f593863e 4e671e62c426f556bf26621b8fb5193c8b4ed36e1d473b66f20b8abd9bdb5fd5 ff1694b11d9547097950b5aca16f37549343ce47ee3d6b0d21a015c6cc23efaa ebf65f85dbd91db3821634f94f30fdafe0d84da2d11f0a75f82c136c28893354 f8fada46416f156e9df40c4065c8d4c4062e48c19433bb949661620f2c40f0fb d7a978608ac910715f41decbf9fd82a95b8ab446a48d2944de3015321ff03d72 680f9ea2382d05fa489064335f6b9abd6c6746537f798b9766e97e53d5453863 79ac18881714b27bf3043bbdae334e7551c50f3793bf44af273f5c890ced0aa6 7c8d3d227dca2766c19d5bc21a525556e746699f5216a289f7e2fbc2cc9601ae 6f4cf34a3147a41b1f321ff42d08e53d6642251ee1e75150bc53abe6bb46ccb4 84bead585d4adfed6051df46857faac8619f3b3f3cd104a415212351de235d9e 4be80de646660b0c5be36e07755afef2a3c0d647361f4ce136e3f859701912f4 24417afb270cc93157383cedc6721e9ea2393124428df075cfd2e67254685d3c c0d5650291d854f074e3d1710c1a7f906ee25ab7611b6a8b79d5961415725bb5 9e08970cd2a46a75d12057c816183f8af70f59fa09b4d97d68f72d63a00cd2d6 a1b30f343943f156bbe853e5635e3f069b78eb700ab8195e0a3631a018d4edd8 5861d091ea6f45b6178685e7d424e701b40d93650cd18969427d4008c8849619 98d1ec2e5029f88000cfb68176ade6a7872ff10f2d3330ff25f3f1e4d0f98534 ef73f66169719db6adf72e959213d8e0cd0dc6ea2b79e8f563ba97c8891ad72c b422c1ee5af1f1b82eabe7230b32171c9bd21a8f78fd1b259a53dafb2f2ea02e 66e4ccc6b2b49098080e0c61cca890abd146e95c7b6b39b57c4915789cd3aee4 5c246234beff3dbfe9823a9035e8ab281aeb29c3d929e6b858a0238c4825 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 14 /slashbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile 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0857d8ce26e80712cc46c2116a092a2f1f6c9ab77ea13f96270524870fe430e4 bbddb65c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 69 /E put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 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c4fde80b75f613b3cd1911767e8b0433f5fcb968bc3ae4e5b39be5370206e1bb 7fe07c47fd7da86aebd0e9d5e2fa1d1fdd2e4a66abefe42f570cfbf17d054a8b e6fbdb1b38232efb3719d4ffbefdf993ffc487b88bc9c1b53fa3706ddd818195 3b700e03ee3378d04c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 22 /mu put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 67 /C put dup 68 /D put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 79 /O put dup 80 /P put dup 82 /R put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 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4ecd13f258507bd8727e35727ea51bd62663f3638fd7050bec62273920412efe abda9a228a3fc124aac0969dac5ed6b5c68c3be15dbb038c5a0fc3a382a1525a 70 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 2 /Theta put dup 3 /Lambda put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 20 /caron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b872c2d3ac4391eedd46c6cae184e1af981f95ce21204dc4babbba38ac cc0a16964fcb3e98501c0392030d486c9aa8b35acadbea5e2bf9b4e732a90a25 6a3227e489f245b82b26f4e485b7567ffff5ee02e89d640e56b11d661791f511 df9cef5821b08b65b671db106348a8bdbbcd377bf82aef6020dfc0cac6fe1932 7140f3fc917d249129ad07ffdd9b6998cf46901131c08bc0e77fa38ee04143bc f4804222bee432c8d0d89ded7fc94b221bc7158766b997de21e51fa6e41af7fc 3c8329a272bbd2dedcf1bd2eb9d8664abc336bb5697c0c5cb95c7d93323c32bf 766f35e038d2d3b4e5c4e39d9923a1e36271d2197ca4a121765a994ecf4dffc5 9e92e783fbabe550dbbe3a6c21fca2652b83d3e550bdfa9b0044f0bb4af8f6d6 b145c7f1ad0e88f63a2385401cccb0e80f36a7454e6685a0199deb1d52a661ac 0244fd9d4de92e28930395a3b8c436bfc939fede4b97e8a1164a68d975534921 c4d36f2591 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 27 /sigma put dup 59 /comma put dup 64 /partialdiff put dup 72 /H put dup 86 /V put dup 108 /l put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe dc6e136cbbbcb3f67509964f9f281ebf81fe5b018122eaf66c4a838487e82e18 6e006093042848a903efb3469ab6b4049767aadb95c30408dfd58d8a10f4cb22 168decd9f3ee100f07b49aa44c92139b669cc312ba20192454eb2375be6284b0 26659d964b96ae82d4942e758027fcf23c25ed01115af27ce7f20efe2a822bb6 84004f20243a49c9e93301fc21b80815c033c3e2ba58ef53da2157d524b395f2 b37abca13bc6a2f42e824ab7e47106176b0d6db267fbb795ac7425582df2e3dc 55863468a9200742bd7b552c48f8cf58bc21343bd3b95abfa140f33f37c6f3f7 8b0d8a5154eb7c1f62ec598267f13e841a3e64172663935ac8b665d86540d316 ddece329c008049c5e74b27d59022c5515059bc3b89370b1bc6a169c888bb325 e0b74282d6f053a50da4024ed1e433271a32ab8c17d41c632b41cf9f3ecd5fe1 24daf7aea7ce8a63047b245822930d517df7baaafa69d2d17f7d93cebd45416d f3459365434123a76ccf883a4973bab19807e1f0c4bab03a45f7ed69ff2660ff 3963e4def11001eec4238c368a39d874fd30b3c14f4186ec7700fbce22abe468 eb32baf7b0164f8f21a9b5e706b91411faaa44467cb2180ab03de375c1ab93e9 76e11ab92fdd4a7280a6d1a8dc65c4c89d04c8ed6988045a2a3de9a4c7b1efd3 75b2966b0ee23a33c1bd122b66e9c8e49bfa1eff0a4497bc9604c3f37277d73b 4afd6af378617c08942430ee0e9f56dfb2c3e27753a0055b1b91ba587348d839 9502bcc9f1936211856d45bf7485616b96200f0d04769054e72ef8a3ff750436 1057b4cf4d7f5775a714cc035b96a1512a35571f0d2ac746f911bb113f661d6f 881ab846275f69a5078c566bef5b5ffb9567c72cd37ea74df4f00cae7deefd42 b4ae00583da128f42d9854f51d59d26d8a2be79f705b1a9029175488325837ab 31371b0c12fd2b7fbe3b26ebf8e1c8afbcb6b81494e8b86c0662e0cc25a3fb6e c590cd881cd864a1b862418230d103dbbc52d01d6d97b8a076145165c32170f8 9428990f233c1b073e9c9bb72ea904707af1e79bc089828f543c1138ea6a7592 cbf4ea05f0bbce7fdac40dd3177b7ea2069d2f3a522e9a0092b4768eb70e3b3a 27f67a6f5fe8ae6df6702092e8cbd05da8888fb1ce83a88182d11e88bc2a29e8 c32a77c73c600f55b5d6e1a6a30e8814101ca7d75b1fd45754c3ad456063d25b 5e012a418fa0900a057368ae2438766c80d85d138009b6d12ffa30da8dd4bcd7 ad20231e71a4d5eaa2e3c24f48d770840026a6c3a3765b3109faaf673f86ee45 3e8926f65b7af8f3625a4085070ec33a872e5b7bc8242f0d9ca58c309f1284d0 2ca4e0e8de9a842642091be944c830392ec3e6c86d667d03fddff5c9a57f995c 7689fb1f32ed7c4cb4ff752668f9ddae2addcd0a01e0e9cf438ecaf02ef094fe 114899029accb68b16560d3488c7e35b4f01d6eb804b6538b961ad11fc167222 574120a52cee3347f8106a37a527f885376cb5914af68be92be0ca48a570114d 01f8c6e8f4588eee682a1d661fc9e4bc9bebcee33d1987882e16b49458f6ae66 04c59bc81c6076656788553d82e7c9cd69d3d1153219671d593e3c41d942264f 9a1991af4bce0395c340641d01b9f74e883baa6af78f78b6914d5417eba0e64b de1874661fc1615d97e4d86df99d0c7a485086efb8b00a1a9b75471dd134d71c 9d171bfff5dc3f5387807f08b95f550110c38d758b345f51e2e39c275e38975e 342d1510e11c462eb1ee8b8a3ce699ef2cca9b48e34fefef9b546d176724446e 0cfc246df652ef33422bcf35c7b443c253eee98d9f2a8dae8cbc8489438c0ca1 acf1b15c2181c241712d41f1e0063f9e3a2db5d7b778a9c3a94bfccaab4a189a c40bf2aad8c929a5c683139893ec80dade887662aaca27daae75ad9ec9eba21a ff05d71485fe915b03a7b22dd9d30da7db3182ba84e71ff932d7f7386c625cfa 6843eb52612ff16f18cbbd967775ed05aa9599e7bc0863548513f974cb89d37a 120945c0ea2b9f6d6f40cb83875575243b185c95aca438d3051af096bac7d8e1 2dcc951755eec73389eb47ff051ba69b3e010367b5a2e3a62831dc3cee1cf498 6a50ea1fdab24db296cdcc385b2e0685c6fa9677baf3ec30ca7ad520a02ad851 4249dd152ebb212272e87a33ba6b64246172f1096bc3f56d5e9b5141f69df443 a8b6cb8e4a3baf28a971d6b1f7ccdee4e74d4f05f97547eb518a318fe9e737e8 adc99e6fb4cf7f3547e77034362b3fd6ae73c3a027ff45bcfcb8030910c22786 85104b3aa8cd68270c159dd412251bb12c48bb260f7d6b440a12faee6066623c 88cc6e0affc09e7e831700b4681acda0461bea66107b8344ca73d94bb615aaf6 4422be3b02ce6dc36c6201d426fb4b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 117 /u put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 a298b221c4a820c700ca964fc99e6b1d9eb0bafc39be9aa9dffa2fa326b2a466 1e67261bb3dbc85844e76f9313ab533ee16bcb478166de5502b68e6ec1ea6110 f53206cf1cfea419571b0243244b32ea38b828a01e62b03bcf476a32c2341cec b8a0fa2a3f3f339315d5f3aa7704408f6617d026ffee4925aa48537a4b2e556d 225be09392546ad561cd46cf766f0ba14c6854c5af1a528c3842b0c5aeefefcf 19f19cbce5d967153c4f88e51c776cfbe10d39328466f7a3b83d85601ef4d1ce 0cc136610fa6a74d9676e4e99291d565cad27b9eb2337b0eba4148efd0eb2866 ebf0a4209ba135ad1376cf9cc2d4110f36121055caa3f2e639889580b9ff1c21 9f2c9bfa30dd03355c423cf85e4aa9c03527d6cdce3f5940317afa73db196e17 ebc90d05b78247e4649cb59fe65d2291f33d4b57faf1e03f142912d8dc827daf 68a0ee58c70ede71f18f00a7a8f8d23d93f50b8c3459a3c5c503df474aea60bb 0412359c1051e97937ed1ca4fe67d3ccb59670b59f85c0eff853882038e387f4 a7005817388d83734be3e2b0105af553713485dc63249c5c7ed095fafd67fc7b 1402610c2b993ac4ec93eb6364008cc15896de9881737c0e0d5dae03c5ffdfcd 8f4720de919589cdc48fbdf0d4b0bf39efe6b1bceff8898e2dfe280e8977bc88 74f4bc5ba6cb407312724aec619119a06e557e38b235f0aa58c843789496cd45 005ce278d9a2facafe3c58d00fe16ca12a3a3a79affa32762a4e76e3f9e2de84 7b7366b13126408908a0ccdbefa060b94f67f3cc99ba21899bd98c799c48d5b4 d3f7435aefdc22be42bed261bf7ef0e9c45bedb561d18cebd24bf3bc32782069 e5ad167ca10dad4942754779f7360259ec4f6fed4c920acb2f78d4e356aee61f 5db9ffdbe3a27f832fcafa397f665a2687b126584dcf7f76b65effd387bf571a 005ad52260d6df08d185e66d077aaf98e20ce8771b0068af46422d4fb3d2c7bc 62da60842baee5e6509db52219e34b9c734641a75a1a5cb4c2d26e70f1034b1e cedd35dddb35519aceee97277553690be0417fc82e1c160e40cbaa7d24bea1c1 d7b8336e573ad25548801af5e402df7041f62ec96d665bf3527f64200f452a02 27aa6dcbb16a6ba4059ad6a7e95a4c8d185414a5914df3e5060029dfb99d199e e1c9f2d93c0ae8005ab1576c6427124b1676616264bab60c4dcba74fd0b6f208 e1de5666473486bb4dab7d587bc73324fdf93797c8b670487873331a4ef7e68d 72f75f94afff84854db67bc69872d8131870c7e63bfa98fb3bb23e093e92f4f2 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1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 72 /H put dup 82 /R put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 54 /six put dup 56 /eight put dup 61 /equal put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 72 /H put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 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cf0fd6d8a168f6292505a6e6c189f95b583b6975f2f91a05cf671e146aeb9b0d d43b4346f6671a65e36f1d657fc5048afa24900cbe6d442b7b680edbc8f2a87d ee9f6f94a97fae7fb4ac414590e1ba830affcba3532d08c0d970f8893746b42b 66286277e6a8cea72e9c57dbea131771e0d377826256cb89390a8012cb0e53ac d5203f7305ca92cd75c2d8966ed4b8f347f440d73efb25699b9028a821b7c987 7cc10df89484e0cbdc64c187c74c2bae545ebf8e8cb26a7944e3225f9f9c4a9d 4bd493ff61375b1d9fb40a35ee85d1aceb1d37db2dcb489b10b1242e1d09b253 32b29a01c343585f58c8f441c02fab85ab2327b29ba7bd44afdfda3cb22b7fe6 0676d0d5dbc170dfa0e2dda59126176a01d89b06cb2afe33c2ea29f10249ff9c 3113e4e2400bd7f58ff7dd3d5795a1762fc300ff986a05341001ffd7b01111e1 c7d2c3707bcff0a979a956d93a03955dd5c22b28c10838fa6706e478cae218a5 8d2d5962108dd871c5a26be01f18b0721d74e7891f7ba8ec5cb026ec7aecc07f e1474fe57777af16b4d4bccead5c013646f99f07985a6ac74415f24e58bcb744 9e444e6bedbddfb025e4a699a93a9970cc82e072d953e79c26ba15d9678d678f 8b49f328d37b4b0a9e2fb30415bb9acd618c9c18dcb50e794f3dfc026669231e deb89c201eb0dfeb2c6de304c9039093cd8d09c950472b671ee1aec17a7807d0 a3037bfdc49b231d78387c45a568c790ef3fa5370bbcac0e3ba02faa1b81df84 a8bb3d0a41c39849c204c46d73db83e0a90751f61f0fc1fe5b3a6816c87963c0 4a96f6cd04577ecf9d5510811fa3a117a53e33358bef68e5d5b1cdf27873e124 8814f00e1ace27ae0a519de2d2efbf43faf30273db3e4b1f3951eb0373b86b40 ce12b21e82c89ba6ecfb9e861a71421bf9b2956943dee9a11ea19fd2f33c2800 e5460f9b55786bc756b6c2b8b82e590016fd244e3f23f63bcdf46cf72b5d5bba b91694205f9e46c1878aa9f2d07f19db3f201c4e08934926b4b9084499bff870 88c29c2406f62d4cdba8807763b22ec433ecdd20d299c81b22bbb6e10c99572b 666486f249c462dacf4f06551cf150e8b2029c458f5a684d0a61b358ddb5dd1c 4ae9361c4109aa74679fa568f5c0e7308f8144aff8b0a808 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a265296b02b4e4b35884f985d29521ee7327887d983bb92dd84ff646661 2b348b5e558d3bc65047afcb202750ea6e9c18a769a4e945e0954787ee65948e 7e8e443a9f2b9f5408e0febf523423b2200166ade223570bd17a6f7dd0386f3d 2246ebb40648548e0a3ff7bab6637a16137c24b042010a3cce2ba9afb9317fcf 4a5b893c4915859ff2b53a77ff0334966f7a234ea8a4349c3e048acd6876f5f3 17b9c238f37ed4231b5e8b661ba52c32821de47f7a0f6fa9c7e0f4bf09bf98a1 8008087aa44eaf37894e83891f552626f1f46563b5188ad949521e1a398eeba3 b9b8f4ac3c24785d50bdf01e5c9b7e4ea1d6b7b6c8019d2d5f353cacf216e30b 71f09d9b95805210964ccbf8f2e30aa04ac3a3a57bde635c9d2be1f5c8f64ad6 19764c72aca8152a51003bc396bce21a56c29306bccc36a37d816de2568e7794 a8f80d857132a3dbaf3ef75409cace85e1b85a8233ef2757a507158bbba8f5f6 d9c08ab4ee39f33a1c1a41ff2b8e81320050b6c4c167596b93918824a65fe1b0 6e7dcf5d7e128a77e5509f3a3a2af6f71de243cccfa5788088d277175e59bd61 a3b5cdc10cfd232cc14682d3b37d3aca68df524af6f38511e5627286034a7c08 1c7596bfc3adecc7d1868cb9d645f8cb61041bccad92249142542934977000c1 bfbbaf570bf1a3fad2368d1e6480ed485642acd1d44258a885562397035dc979 b08877eadba7c99963e7d866be89270b10629964b5c88439c08a7721e2fd6a7d 2a03d9cf2d7fa713411824db51a9141a7c3ca37feeb758ab37376d75b9ad8534 6fe75968a1e7f6b9b549b7bbef5da78116a7d029e6e95db78593abf789ab985e 301f5945c1628f591b4867079526cd9c779b05cd30de4d9dbb820ce964cb3acf cf693b39b15335996a208373a9a1cd362e93d740535d4fe06aaef81a77034010 883c098f8fe19617d6a3cc3ccaa63742d3b251b8baa94aba7ebb7f3cad53c3c5 2c626e3918191af1e343757e0fc5f9fa620ce7e655482aaab7303710167e3041 148e3e6aae44ebe904d96a7dea9281fdbdf93b03ccd08a315df83531214f1726 dd6eff4d5f494262983201ee2afd6cc77eac76155fc59d9e42a5c45735a37af6 c3c9c4aa7c535717838a1ce8ca3424188a3ddc2bb7f58483ef08fe9b6ea090c8 bd5a639bb56a54bed34a443f0f1624f13fffd6b4f24024573a6cdc110f2a85b3 f669c706b0bcf205ee9ff8342b38525e9cbf1f1b772a71c7718b30bd59f46490 dd92c121c9fd0b99edf33c57611ce19ca35fe17079c90ce99afdfd9227184562 f4e2dc14aea6d0989a8f01f17c0983ae121614d8f578d34ead9c05a88007bc4f 90e135d97abf57c1e32fd94ed63ceeddab6c1d42f3238e6cf85a063d1239a1d9 c5de6f9e7954b873ccef98d8b44f1761e66da28b9759a24f181c9f7ca8ca190d 4d5a52b1286ad43b5f8caebe64283da13e6a9e66536a821ebc1b50e98073d79e 335c244c2b8efcd7c0b223983db1608d6f1837ae6f73d970e567960e7e03d1a6 5f7dcaeecd4f9afd0266592ed051f800ee3d5a8954460f3f57a9fb9716f4dca2 616d5e77e19918f2b1c290fa9d049267df21a4b33a06e29c758e0ecac9746be0 b7988da1634f2e7162c2c483f1c8d78eaefc94e907d3bcc79cc389979ab0dc15 bf1b2371fe907e9c6f46e26260566e38ed55175ce116e4da7024db03b6657b44 1543e575e850440d7613b96baa8320e2cde5ab834d1f333a9a749c3ab079546c 3429c90c61bfda4b758861c2e351866a18c4d14d3bee4d863edce298231a71dc 9e4c701ac2d01fead65c2244a81865874b1c22125e9f11720450c10366eb6eec 694ba3acd45ea6f9cda354cc938222a523494fa570ba76d359afd28f6ca82c35 dc9890f6e09346d93734ef8c82d7b62344fb8103a1fe6f84fb56f55e1969c4bb f61c916f5fc7aaf919d620ed2c25d2bac648ede51f981afe1d4ecf8263830258 75069803a3032b0fba0522 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 83 /S put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueID 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 40258431 52099146 1000 600 600 (surf-bulk.dvi) @start /Fa 175[83 80[{}1 119.552 /CMSY10 rf /Fb 205[74 50[{}1 99.6264 /LASY10 rf /Fc 253[134 2[{}1 172.188 /CMSY10 rf /Fd 173[82 15[88 66[{}2 99.6264 /EUFM10 rf /Fe 147[20 7[32 100[{}2 49.8132 /CMMI6 rf /Ff 165[44 5[44 1[48 14[48 67[{}4 66.4176 /msbm8 rf /Fg 128[50 5[47 45 65 45 52 32 40 41 45 50 50 55 80 25 1[30 30 50 45 30 45 50 45 45 50 4[50 4[97 72 1[70 55 71 1[66 75 72 87 61 1[51 38 72 1[64 66 74 70 1[72 7[50 3[50 3[50 2[30 35 30 2[40 40 5[50 14[50 7[60 11[{}55 99.6264 /CMTI12 rf /Fh 134[59 1[81 2[44 44 46 1[62 56 62 93 31 59 1[31 62 56 34 51 62 50 1[54 12[78 1[84 1[77 1[88 1[67 6[74 1[81 1[85 7[56 56 56 56 56 56 56 56 56 56 1[31 46[{}38 99.6264 /CMBX12 rf /Fi 133[60 71 1[97 1[75 52 53 55 1[75 67 75 112 37 71 1[37 75 67 41 61 75 60 75 65 12[94 75 100 1[92 3[81 23[67 67 67 67 2[37 46[{}32 119.552 /CMBX12 rf /Fj 140[24 15[27 48[30 30 30 4[47 43[{}6 49.8132 /CMR6 rf /Fk 207[18 41[48 5[48{}3 49.8132 /CMSY6 rf /Fl 145[35 3[20 2[35 35 18[43 10[60 8[55 8[0 3[47 71 19 14[71 12[55 12[55 55 4[20 55{}17 66.4176 /CMSY8 rf /Fm 143[100 21[55 1[144 5[47 1[105 19[89 1[89 1[89 15[80 12[75 75 6[73 73 59 59 1[58 1[33 2[58 58 6[46 46{}21 99.6264 /CMEX10 rf /Fn 165[66 5[66 1[72 12[66 1[72 67[{}5 99.6264 /MSBM10 rf /Fo 143[83 1[50 2[50 28 39 39 50 50 9[66 7[54 60 84 1[69 1[82 5[84 1[72 1[77 2[80 1[77 3[50 2[55 1[0 3[66 100 15[100 3[100 100 1[77 1[77 2[77 77 13[77 3[77 28 77{}35 99.6264 /CMSY10 rf /Fp 133[45 48 55 70 1[56 35 46 1[43 49 1[58 85 29 51 40 1[56 47 48 45 51 42 41 51 41 7[81 1[57 66 57 1[74 1[63 75 2[66 83 1[43 81 3[81 70 1[73 51 1[76 49 76 27 27 25[63 61 58 1[42 55 1[55 2[59 57 5[39 43 51 1[62 11[{}55 99.6264 /CMMI12 rf /Fq 136[117 1[90 63 64 66 1[90 81 90 134 45 2[45 2[49 74 90 72 1[78 13[90 120 8[60 4[124 16[81 81 81 49[{}23 143.462 /CMBX12 rf /Fr 128[49 49 3[43 51 51 70 51 54 38 38 38 51 54 49 54 81 27 51 30 27 54 49 30 43 54 43 54 49 2[49 27 49 27 1[73 73 100 73 73 70 54 72 1[66 76 73 89 61 76 50 35 73 77 64 66 75 70 69 73 3[76 1[27 27 49 49 49 49 49 49 49 49 49 49 1[27 33 27 76 1[38 38 27 4[49 13[49 49 4[81 1[54 57 70 76 5[68 76 1[61{}87 99.6264 /CMR12 rf /Fs 252[34 3[{}1 58.1154 /CMSY7 rf /Ft 147[27 21[53 13[76 7[48 36[52 27[{}5 90.9091 /CMMI10 rf /Fu 253[71 2[{}1 90.9091 /CMSY10 rf /Fv 129[35 8[39 1[28 28 1[39 35 5[20 3[31 39 31 39 4[20 1[20 29[55 8[35 35 35 35 35 4[55 1[27 27 37[55 2[{}23 66.4176 /CMR8 rf /Fw 133[33 35 40 51 1[41 25 33 1[32 2[43 62 21 37 29 4[33 36 31 30 37 12[41 1[53 9[58 9[55 35 1[20 27[44 42 1[31 12[29 31 36 1[45 11[{}31 66.4176 /CMMI8 rf /Fx 165[61 90[{}1 90.9091 /MSBM10 rf /Fy 133[40 48 48 66 48 51 35 36 36 48 51 45 51 76 25 2[25 51 45 28 40 51 40 51 45 9[93 2[66 4[71 1[83 57 3[68 3[69 2[68 3[71 14[25 30 25 71 1[35 35 40[{}39 90.9091 /CMR10 rf /Fz 139[41 41 43 14[46 58 51 31[79 65[{}7 90.9091 /CMBX10 rf /FA 196[23 59[{}1 83.022 /CMMI10 rf /FB 249[65 2[42 3[{}2 83.022 /CMSY10 rf /FC 134[44 1[60 44 46 32 33 33 1[46 42 46 69 23 2[23 46 42 25 37 46 37 1[42 11[62 1[46 3[65 62 76 1[65 5[57 63 60 1[62 8[42 1[42 42 1[42 42 42 4[23 44[{}37 83.022 /CMR10 rf /FD 134[62 1[85 62 65 46 46 46 1[65 59 65 98 33 62 36 33 65 59 36 52 65 52 65 59 7[88 2[88 88 85 65 86 1[80 1[88 107 73 91 60 42 88 2[80 89 85 83 88 7[59 59 2[59 59 59 59 59 59 1[33 1[33 23[59 59 8[85 10[{}55 119.552 /CMR12 rf /FE 138[88 61 62 61 1[88 1[88 7[47 70 88 70 1[79 13[88 83[{}12 172.188 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%EndSetup %%Page: 1 1 1 0 bop 962 809 a FE(Surface)52 b(States)g(and)g(Sp)t(ectra)1005 1146 y FD(V)-10 b(o)7 b(jk)-7 b(an)37 b(Jak)-6 b(\024)-53 b(si)m(\023)e(c)1714 1102 y FC(1)r FB(\003)1836 1146 y FD(and)39 b(Y)-10 b(oram)37 b(Last)2665 1102 y FC(2)p FA(;)p FC(3)1144 1401 y(1)1190 1445 y FD(Departmen)m(t)g(of)h (Mathematics)1233 1594 y(Johns)h(Hopkins)f(Univ)m(ersit)m(y)852 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b(Our)36 b(next)h(results)h(iden)m(ti\014es)h(their)e(ranges.)56 b(W)-8 b(e)37 b(adopt)f(the)h(shorthand)g Fo(h)p Fp(t)p Fo(i)d Fr(=)g(\(1)25 b(+)f Fp(t)3620 1126 y Fv(2)3660 1163 y Fr(\))3708 1099 y Fj(1)p 3708 1111 31 3 v 3708 1152 a(2)3752 1163 y Fr(.)0 1283 y(Let)902 1463 y Fo(T)53 b Fr(:=)1140 1382 y Fm(\010)1198 1463 y Fp( )32 b Fr(:)27 b Fo(8)p Fp(R)q(;)17 b(k)31 b Fo(\025)d Fr(0)p Fp(;)50 b Fo(k)p Fh(1)1940 1478 y Fw(R)1998 1463 y Fr(e)2041 1421 y Fl(\000)p Fv(i)p Fw(tH)2208 1463 y Fp( )t Fo(k)28 b Fr(=)f Fp(O)s Fr(\()p Fo(h)p Fp(t)p Fo(i)2685 1421 y Fl(\000)p Fw(k)2782 1463 y Fr(\))2820 1382 y Fm(\011)948 1761 y Fo(F)37 b Fr(:=)1188 1621 y Fm(\032)1263 1761 y Fp( )31 b Fr(:)d Fo(8)p Fp(R)h Fo(\025)f Fr(0)p Fp(;)1801 1626 y Fm(Z)1857 1851 y Ff(R)1925 1761 y Fo(k)p Fh(1)2031 1776 y Fw(R)2089 1761 y Fr(e)2132 1720 y Fl(\000)p Fv(i)p Fw(tH)2299 1761 y Fp( )t Fo(k)2416 1720 y Fv(2)2456 1761 y Fr(d)p Fp(t)g(<)f Fo(1)2776 1621 y Fm(\033)2851 1761 y Fp(:)0 2018 y Fr(Clearly)-8 b(,)34 b Fo(T)58 b Fr(and)32 b Fo(F)42 b Fr(are)33 b(linear)g(spaces)h(and)f Fo(T)53 b(\032)28 b(F)10 b Fr(.)0 2322 y Fh(Theorem)75 b(1.2)49 b Fr(Ran)17 b(\012)966 2286 y Fl(\006)1053 2322 y Fr(=)p 1156 2242 80 4 v 27 w Fo(T)54 b Fr(=)p 1367 2242 82 4 v 27 w Fo(F)10 b Fg(.)45 b(Mor)-5 b(e)g(over,)34 b Fr(Ran\012)2227 2286 y Fl(\006)2314 2322 y Fo(\032)28 b(H)2503 2337 y Fv(s)2535 2322 y Fr(\()p Fp(H)8 b Fr(\))2700 2286 y Fl(?)2759 2322 y Fg(.)0 2551 y Fr(W)-8 b(e)31 b(recall)h(that)f(the)g(w)m(a)m(v)m (e)i(op)s(erators)e(\012)1538 2514 y Fl(\006)1628 2551 y Fr(are)g(complete)h(on)f(a)g(Borel)g(set)h(\002)27 b Fo(\032)h Fn(R)j Fr(if)g(Ran)p Fh(1)3451 2566 y Fv(\002)3510 2551 y Fr(\()p Fp(H)8 b Fr(\))27 b Fo(\032)0 2671 y Fr(Ran\012)245 2635 y Fl(\006)304 2671 y Fr(.)44 b(Therefore,)33 b(Theorem)h(1.2)f (yields:)0 2884 y Fh(Corollary)38 b(1.3)49 b Fg(If)34 b(the)h(wave)f(op)-5 b(er)g(ators)35 b Fr(\012)1687 2848 y Fl(\006)1781 2884 y Fg(ar)-5 b(e)35 b(c)-5 b(omplete)34 b(on)g Fr(\002)p Fg(,)h(then)f Fp(\033)2899 2899 y Fv(s)2932 2884 y Fr(\()p Fp(H)8 b Fr(\))21 b Fo(\\)i Fr(\002)k(=)h Fo(;)p Fg(.)0 3113 y Fr(In)33 b([JL1])f(w)m(e)i(ha)m(v)m(e)f(in)m(tro)s (duced)h(the)f(notion)f(of)g(resonan)m(t)h(sp)s(ectrum,)h Fd(R)p Fr(\()p Fp(H)8 b Fr(\),)32 b(and)h(w)m(e)g(ha)m(v)m(e)h(sho)m (wn)0 3233 y(that)27 b(the)h(w)m(a)m(v)m(e)h(op)s(erators)e(\012)1098 3197 y Fl(\006)1184 3233 y Fr(are)g(complete)i(on)e Fp(\033)t Fr(\()p Fp(H)8 b Fr(\))j Fo(n)g Fd(R)p Fr(\()p Fp(H)d Fr(\).)41 b(Th)m(us,)30 b Fp(\033)2814 3248 y Fv(s)2846 3233 y Fr(\()p Fp(H)8 b Fr(\))28 b Fo(\032)g Fd(R)p Fr(\()p Fp(H)8 b Fr(\).)41 b(V)-8 b(arious)0 3354 y(estimates)34 b(on)f(the)g(lo)s(cation)f(of)g(the)h(set)g Fd(R)p Fr(\()p Fp(H)8 b Fr(\))33 b(are)f(giv)m(en)i(in)f([JL1].)0 3686 y Fi(1.2)135 b(Random)46 b(b)t(oundary)e(p)t(oten)l(tials)0 3916 y Fr(Let)33 b(\012)g(b)s(e)g(the)g(set)g(of)f(all)g(b)s(oundary)h (p)s(oten)m(tials,)1520 4120 y(\012)28 b(=)g Fn(R)1794 4079 y Ff(Z)1838 4056 y Fe(d)1906 4120 y Fr(=)g Fc(\002)2036 4205 y Ff(Z)2080 4186 y Fe(d)2160 4120 y Fn(R)p Fp(;)0 4382 y Fr(let)33 b Fd(B)f Fr(b)s(e)h(the)g(Borel)g Fp(\033)t Fr(-algebra)f(in)h(\012)g(and)f(d)p Fp(P)47 b Fr(a)32 b(probabilit)m(y)h(measure)h(on)f(\(\012)p Fp(;)17 b Fd(B)p Fr(\))32 b(of)h(the)g(form)1614 4600 y(d)p Fp(P)41 b Fr(=)27 b Fc(\002)1902 4684 y Ff(Z)1946 4665 y Fe(d)2026 4600 y Fr(d)p Fp(\026;)0 4861 y Fr(where)43 b(d)p Fp(\026)g Fr(=)g Fp(p)p Fr(\()p Fp(x)p Fr(\)d)p Fp(x)f Fr(is)h(an)e(absolutely)i (con)m(tin)m(uous)g(\(w.r.t.)72 b(Leb)s(esgue)43 b(measure\))f (probabilit)m(y)0 4982 y(measure)35 b(on)f Fn(R)p Fr(.)49 b(In)34 b(what)g(follo)m(ws)h(w)m(e)g(use)g(the)g(sym)m(b)s(ol)g Fn(E)p Fr(\()p Fp(f)11 b Fr(\))35 b(for)e(the)i(exp)s(ectation)g(of)f (a)f(random)0 5102 y(v)-5 b(ariable)33 b Fp(f)43 b Fr(and)33 b(assume)h(that)e Fp(d)22 b Fr(+)g(1)27 b(=)h(2.)0 5268 y(The)34 b(main)e(result)i(of)e(this)h(pap)s(er)g(is:)p eop %%Page: 5 5 5 4 bop 3731 100 a Fr(5)0 407 y Fh(Theorem)75 b(1.4)49 b Fg(L)-5 b(et)28 b Fp(U)931 422 y Fv(p)r(er)1064 407 y Fg(b)-5 b(e)27 b(a)h(p)-5 b(erio)g(dic)26 b(b)-5 b(oundary)28 b(p)-5 b(otential)27 b(and)g Fp(H)35 b Fr(=)28 b Fp(H)2898 422 y Fv(0)2944 407 y Fr(+)6 b Fp(U)3092 422 y Fv(p)r(er)3202 407 y Fr(+)g Fp(\025V)22 b Fg(,)29 b Fp(V)49 b Fo(2)28 b Fr(\012)p Fg(,)0 527 y(wher)-5 b(e)39 b Fp(\025)g Fg(is)g(a)g(r)-5 b(e)g(al)39 b(c)-5 b(onstant.)58 b(Assume)40 b(that)f Fo(h)p Fp(x)p Fo(i)1911 491 y Fw(\013)1961 527 y Fp(p)p Fr(\()p Fp(x)p Fr(\))d Fo(2)h Fp(L)2346 491 y Fv(1)2385 527 y Fr(\()p Fn(R)p Fr(\))26 b Fo(\\)g Fp(L)2717 491 y Fl(1)2792 527 y Fr(\()p Fn(R)p Fr(\))39 b Fg(for)g(some)g Fp(\013)d(>)g Fr(2)p Fp(=)p Fr(3)p Fg(.)0 648 y(Then)e(ther)-5 b(e)35 b(is)g(a)f(c)-5 b(onstant)35 b Fr(\003)27 b Fp(>)h Fr(0)34 b Fg(such)h(that)g(for)g Fo(j)p Fp(\025)p Fo(j)27 b Fp(>)g Fr(\003)p Fg(,)35 b Fp(R)29 b Fo(\025)f Fr(0)34 b Fg(and)h Fp( )c Fo(2)d(H)q Fg(,)892 920 y Fr(lim)864 982 y Fw(T)10 b Fl(!1)1118 852 y Fr(1)p 1082 897 120 4 v 1082 988 a(2)p Fp(T)1229 784 y Fm(Z)1328 810 y Fw(T)1284 1010 y Fl(\000)p Fw(T)1410 920 y Fn(E)1493 839 y Fm(\000)1539 920 y Fo(k)p Fh(1)1645 935 y Fw(R)1703 920 y Fr(e)1746 878 y Fl(\000)p Fv(i)p Fw(tH)1913 920 y Fh(1)1969 935 y Fv(\()p Fl(\000)p Fw(c)2082 947 y Fe(d)2118 935 y Fw(;c)2169 947 y Fe(d)2205 935 y Fv(\))2236 920 y Fr(\()p Fp(H)e Fr(\))p Fp( )t Fo(k)2518 878 y Fv(2)2557 839 y Fm(\001)2619 920 y Fr(d)p Fp(t)28 b Fr(=)g(0)p Fp(:)663 b Fr(\(1.5\))0 1184 y Fg(In)34 b(p)-5 b(articular,)35 b(for)g Fo(j)p Fp(\025)p Fo(j)27 b Fp(>)g Fr(\003)p Fg(,)1163 1446 y Fp(\033)1218 1461 y Fv(s)1250 1446 y Fr(\()p Fp(H)8 b Fr(\))22 b Fo(\\)g Fr(\()p Fo(\000)p Fp(c)1682 1461 y Fw(d)1723 1446 y Fp(;)17 b(c)1809 1461 y Fw(d)1849 1446 y Fr(\))28 b(=)f Fo(;)199 b Fp(P)36 b Fo(\000)23 b Fp(a:s:)962 b Fr(\(1.6\))0 1880 y Fh(Remark)39 b(1.)47 b Fr(That)34 b(\(1.5\))f(implies)i(\(1.6\))f(is)g(seen)h(as)f(follo)m(ws.)47 b(Let)34 b Fo(N)48 b Fr(b)s(e)34 b(a)f(coun)m(table,)j(dense)f(set)0 2000 y(in)k Fo(H)q Fr(.)63 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b Fo(2)3530 2642 y Fr(~)3519 2667 y(\012)33 b(and)0 2788 y Fp( )f Fo(2)p 189 2708 97 4 v 28 w(N)42 b Fr(=)27 b Fo(H)q Fr(.)0 2908 y Fh(Remark)38 b(2.)43 b Fr(The)34 b(constan)m(t)f Fo(j)p Fp(\025)p Fo(j)f Fr(is)h(a)g(measure)g(of)g(the)g(disorder.)0 3028 y Fh(Remark)43 b(3.)55 b Fr(The)38 b(op)s(erator)e Fp(H)1255 3043 y Fv(0)1319 3028 y Fr(+)25 b Fp(U)1486 3043 y Fv(p)r(er)1627 3028 y Fr(ma)m(y)37 b(ha)m(v)m(e)h(some)g (surface)g(sp)s(ectra)f(on)f Fp(\033)t Fr(\()p Fp(H)3313 3043 y Fv(0)3353 3028 y Fr(\).)55 b(F)-8 b(or)36 b(ex-)0 3149 y(ample,)h(if)e Fo(8)p Fp(n)p Fr(,)i Fp(U)653 3164 y Fv(p)r(er)757 3149 y Fr(\()p Fp(n;)17 b Fr(0\))33 b(=)f Fp(a)k Fr(and)f Fo(j)p Fp(a)p Fo(j)d Fp(>)h Fr(1,)j(then)g Fp(\033)2044 3164 y Fv(s)2076 3149 y Fr(\()p Fp(H)2195 3164 y Fv(0)2259 3149 y Fr(+)23 b Fp(U)2424 3164 y Fv(p)r(er)2529 3149 y Fr(\))32 b(=)h([)p Fo(\000)p Fr(2)p Fp(d;)17 b Fr(2)p Fp(d)p Fr(])23 b(+)h Fp(a)h Fr(+)e Fp(a)3432 3113 y Fl(\000)p Fv(1)3527 3149 y Fr(.)52 b(One)0 3269 y(can)39 b(also)f(sho)m(w)i(that)e(this)h(surface)h(sp)s(ectrum)g(is)f(stable)g (under)g(short-range)g(p)s(erturbations,)h(see)0 3390 y([JMP2)q(].)0 3510 y Fh(Remark)33 b(4.)42 b Fr(The)29 b(tec)m(hnical)g(conditions)g(of)f(Theorem)h(1.4)e(can)i(b)s(e)f (relaxed)h(in)f(sev)m(eral)i(w)m(a)m(ys.)44 b(The)0 3630 y(metho)s(d)38 b(of)e(the)i(pro)s(of)e(allo)m(ws)i(for)f(more)g (general)h(densities)h(\(non-i.i.d.)e(and)h(correlated)f(random)0 3751 y(v)-5 b(ariables\),)32 b(the)e(bac)m(kground)i(p)s(oten)m(tial)e (do)s(es)h(not)f(ha)m(v)m(e)i(to)e(b)s(e)h(p)s(erio)s(dic,)g(etc.)43 b(W)-8 b(e)31 b(opted)g(for)f(con-)0 3871 y(ditions)g(whic)m(h)g(co)m (v)m(er)h(most)e(ph)m(ysically)i(in)m(teresting)g(situations,)g(while)f (allo)m(wing)f(for)g(a)f(tec)m(hnically)0 3992 y(simple)34 b(exp)s(osition)g(of)e(the)h(results)h(and)e(the)h(pro)s(ofs.)0 4157 y(If)40 b(the)g(bac)m(kground)h(p)s(oten)m(tial)f Fp(U)1298 4172 y Fv(p)r(er)1442 4157 y Fr(is)h(equal)f(to)g(zero,)i 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Fw(T)1284 5094 y Fl(\000)p Fw(T)1410 5004 y Fn(E)1493 4923 y Fm(\000)1539 5004 y Fo(k)p Fh(1)1645 5019 y Fw(R)1703 5004 y Fr(e)1746 4963 y Fl(\000)p Fv(i)p Fw(tH)1913 5004 y Fh(1)1969 5019 y Fv(\()p Fl(\000)p Fw(c)2082 5031 y Fe(d)2118 5019 y Fw(;c)2169 5031 y Fe(d)2205 5019 y Fv(\))2236 5004 y Fr(\()p Fp(H)e Fr(\))p Fp( )t Fo(k)2518 4963 y Fv(2)2557 4923 y Fm(\001)2619 5004 y Fr(d)p Fp(t)28 b Fr(=)g(0)p Fp(:)0 5268 y Fg(In)34 b(p)-5 b(articular,)35 b(for)g Fo(j)p Fp(\025)p Fo(j)27 b Fp(<)g Fr(\003)p Fg(,)35 b Fp(\033)1190 5283 y Fv(s)1222 5268 y Fr(\()p Fp(H)8 b Fr(\))22 b Fo(\\)g Fr(\()p Fo(\000)p Fp(c)1654 5283 y Fw(d)1695 5268 y Fp(;)17 b(c)1781 5283 y Fw(d)1821 5268 y Fr(\))28 b(=)f Fo(;)35 b Fg(P-a.s.)p eop %%Page: 6 6 6 5 bop 3731 100 a Fr(6)0 407 y Fh(Remark)38 b(1.)44 b Fr(If)32 b(\003)p Fo(k)p Fp(V)21 b Fo(k)28 b(\024)g Fr(1)33 b Fp(P)14 b Fr(-a.s.,)32 b(then)h(for)f Fo(j)p Fp(\025)p Fo(j)27 b Fp(<)h Fr(\003,)k(the)h(w)m(a)m(v)m(e)i(op)s (erators)d(\(1.4\))h(are)f(complete)0 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b(of)g(Theorem)h(1.1)0 4183 y Fr(The)34 b(follo)m(wing)e(result)i(will) f(b)s(e)g(used)h(on)e(sev)m(eral)i(o)s(ccasions)g(in)f(this)g(pap)s (er.)0 4403 y Fh(Lemma)39 b(2.1)49 b Fg(L)-5 b(et)35 b Fp(A)g Fg(b)-5 b(e)35 b(a)g(b)-5 b(ounde)g(d)34 b(self-adjoint)f(op) -5 b(er)g(ator)35 b(on)f Fo(H)q Fg(.)45 b(Then,)34 b(for)h(any)g Fp( )c Fo(2)d(H)q Fg(,)783 4618 y Fr(1)p 747 4663 120 4 v 747 4754 a(2)p Fp(T)894 4550 y Fm(Z)993 4576 y Fw(T)949 4775 y Fl(\000)p Fw(T)1075 4685 y Fo(k)p Fp(A)p Fr(e)1241 4644 y Fl(\000)p Fv(i)p Fw(tH)1409 4685 y Fp( )t Fo(k)1526 4644 y Fv(2)1565 4685 y Fr(d)p Fp(t)g Fo(\024)g Fp(C)7 b(\017)17 b Fr(max)1984 4745 y Fl(\006)2119 4550 y Fm(Z)2174 4775 y Ff(R)2243 4685 y Fo(k)p Fp(A)p Fo(R)p Fr(\()p Fp(e)22 b Fo(\006)h Fr(i)p Fp(\017)p Fr(\))p Fp( )t Fo(k)2876 4644 y Fv(2)2916 4685 y Fr(d)p Fp(e;)0 4969 y Fg(wher)-5 b(e)34 b Fp(\017)28 b Fr(=)g Fp(T)517 4933 y Fl(\000)p Fv(1)646 4969 y Fg(and)34 b Fp(C)h Fr(=)27 b(e)1086 4933 y Fv(2)1126 4969 y Fp(=)p Fr(2)p Fp(\031)t Fg(.)p eop %%Page: 8 8 8 7 bop 3731 100 a Fr(8)0 407 y Fh(Pro)s(of.)65 b Fr(The)33 b(result)h(follo)m(ws)f(from)g(the)g(w)m(ell-kno)m(wn)h(iden)m(tit)m(y) h(\(see,)e(e.g.,)g([RS],)g(Section)h(XI)s(I)s(I.7\))397 675 y(2)p Fp(\031)521 540 y Fm(Z)577 765 y Ff(R)645 675 y Fr(e)688 634 y Fl(\000)p Fv(2)p Fw(\017)p Fl(j)p Fw(t)p Fl(j)876 675 y Fo(k)p Fp(A)p Fr(e)1042 634 y Fl(\000)p Fv(i)p Fw(tH)1210 675 y Fp( )t Fo(k)1327 634 y Fv(2)1366 675 y Fr(d)p Fp(t)28 b Fr(=)1587 540 y Fm(Z)1642 765 y Ff(R)1711 595 y Fm(\000)1756 675 y Fo(k)p Fp(A)p Fo(R)p Fr(\()p Fp(e)23 b Fr(+)f(i)p Fp(\017)p Fr(\))p Fp( )t Fo(k)2388 634 y Fv(2)2450 675 y Fr(+)g Fo(k)p Fp(A)p Fo(R)p Fr(\()p Fp(e)h Fo(\000)f Fr(i)p Fp(\017)p Fr(\))p Fp( )t Fo(k)3181 634 y Fv(2)3221 595 y Fm(\001)3283 675 y Fr(d)p Fp(e)0 942 y Fr(and)33 b(the)g(estimate)850 1138 y(1)p 815 1182 120 4 v 815 1273 a(2)p Fp(T)961 1069 y Fm(Z)1060 1096 y Fw(T)1016 1295 y Fl(\000)p Fw(T)1143 1205 y Fo(k)p Fp(A)p Fr(e)1309 1164 y Fl(\000)p Fv(i)p Fw(tH)1476 1205 y Fp( )t Fo(k)1593 1164 y Fv(2)1632 1205 y Fr(d)p Fp(t)c Fo(\024)1865 1138 y Fr(e)1908 1101 y Fv(2)1948 1138 y Fp(\017)p 1865 1182 123 4 v 1901 1273 a Fr(2)2013 1069 y Fm(Z)2069 1295 y Ff(R)2138 1205 y Fr(e)2181 1164 y Fl(\000)p Fv(2)p Fw(\017)p Fl(j)p Fw(t)p Fl(j)2368 1205 y Fo(k)p Fp(A)p Fr(e)2534 1164 y Fl(\000)p Fv(i)p Fw(tH)2702 1205 y Fp( )t Fo(k)2819 1164 y Fv(2)2858 1205 y Fr(d)p Fp(t;)0 1489 y Fr(where)34 b Fp(\017)28 b Fr(=)f Fp(T)523 1453 y Fl(\000)p Fv(1)617 1489 y Fr(.)44 b Fb(2)0 1709 y Fr(The)c(follo)m(wing)g(lemma)g(is)f(an)g(immediate)i (consequence)h(of)d(Prop)s(osition)g(3.1)g(of)g([JL1].)63 b(W)-8 b(e)39 b(use)0 1829 y(the)33 b(shorthand)g Fo(R)713 1844 y Fv(0)753 1829 y Fr(\()p Fp(z)t Fr(\))28 b(=)g(\()p Fp(H)1129 1844 y Fv(0)1190 1829 y Fo(\000)23 b Fp(z)t Fr(\))1377 1793 y Fl(\000)p Fv(1)1472 1829 y Fr(.)0 2049 y Fh(Lemma)39 b(2.2)49 b Fg(If)34 b Fr([)p Fp(a;)17 b(b)p Fr(])23 b Fo(\\)g Fp(\033)t Fr(\()p Fp(H)1171 2064 y Fv(0)1210 2049 y Fr(\))28 b(=)f Fo(;)35 b Fg(then)f(ther)-5 b(e)35 b(exists)g Fp(\015)d(>)c Fr(0)35 b Fg(such)f(that)1221 2269 y Fr(sup)1122 2355 y Fw(e)p Fl(2)p Fv([)p Fw(a;b)p Fv(])p Fw(;\017)p Fl(6)p Fv(=0)1484 2269 y Fo(k)p Fh(1)p 1590 2241 58 3 v 26 x Fw(R)1647 2269 y Fo(R)1731 2284 y Fv(0)1771 2269 y Fr(\()p Fp(e)22 b Fr(+)g(i)p Fp(\017)p Fr(\))p Fh(1)2134 2284 y Fv(0)2174 2269 y Fo(k)28 b(\024)g Fp(C)7 b Fr(e)2477 2228 y Fl(\000)p Fw(\015)t(R)2630 2269 y Fp(:)0 2709 y Fh(Lemma)39 b(2.3)49 b Fg(L)-5 b(et)32 b Fr(\002)f Fg(b)-5 b(e)31 b(a)g(c)-5 b(omp)g(act)31 b(set)g(such)g(that)h Fr(\002)14 b Fo(\\)g Fp(\033)t Fr(\()p Fp(H)2341 2724 y Fv(0)2381 2709 y Fr(\))27 b(=)h Fo(;)p Fg(.)43 b(Then)31 b(ther)-5 b(e)31 b(exist)g(c)-5 b(onstants)0 2829 y Fp(C)35 b(>)27 b Fr(0)35 b Fg(and)f Fp(\015)f(>)27 b Fr(0)35 b Fg(such)g(that)g(for)g(al)5 b(l)34 b Fp(R)29 b Fo(\025)f Fr(0)35 b Fg(and)f Fp(n)28 b Fo(2)g Fn(Z)2195 2793 y Fw(d)2236 2829 y Fg(,)786 3092 y Fr(lim)17 b(sup)868 3172 y Fw(\017)p Fl(!)p Fv(0)1101 3092 y Fo(j)p Fp(\017)p Fo(j)1213 2957 y Fm(Z)1268 3182 y Ff(R)1337 3092 y Fo(k)p Fh(1)p 1443 3064 V 26 x Fw(R)1500 3092 y Fo(R)p Fr(\()p Fp(e)23 b Fr(+)f(i)p Fp(\017)p Fr(\))p Fh(1)1948 3107 y Fv(\002)2008 3092 y Fr(\()p Fp(H)8 b Fr(\))p Fp(\016)2216 3108 y Fv(\()p Fw(n;)p Fv(0\))2372 3092 y Fo(k)2422 3051 y Fv(2)2461 3092 y Fr(d)p Fp(e)28 b Fo(\024)g Fp(C)7 b Fr(e)2813 3051 y Fl(\000)p Fw(\015)t(R)2967 3092 y Fp(:)0 3549 y Fh(Pro)s(of.)79 b Fr(Let)40 b([)p Fp(a;)17 b(b)p Fr(])40 b(b)s(e)g(an)f(in)m(terv)-5 b(al)41 b(suc)m(h)g(that)e(\002)h(is)g(prop)s(erly)g(con)m(tained)h(in) f([)p Fp(a;)17 b(b)p Fr(])40 b(and)g([)p Fp(a;)17 b(b)p Fr(])27 b Fo(\\)0 3669 y Fp(\033)t Fr(\()p Fp(H)178 3684 y Fv(0)217 3669 y Fr(\))h(=)f Fo(;)p Fr(.)44 b(Then)33 b(there)h(exists)g(a)e(constan)m(t)i Fp(C)1825 3684 y Fv(1)1896 3669 y Fr(suc)m(h)h(that)d Fo(8)p Fp(e)c Fo(2)g Fn(R)22 b Fo(n)g Fr([)p Fp(a;)17 b(b)p Fr(],)1064 3889 y(sup)1123 3969 y Fw(\017)1228 3889 y Fo(kR)p Fr(\()p Fp(e)22 b Fr(+)g(i)p Fp(\017)p Fr(\))p Fh(1)1725 3904 y Fv(\002)1785 3889 y Fr(\()p Fp(H)8 b Fr(\))p Fo(k)27 b(\024)h Fp(C)2202 3904 y Fv(1)2241 3889 y Fp(=)p Fr(dist)q(\()p Fp(e;)17 b Fr(\002\))p Fp(:)0 4156 y Fr(Th)m(us,)34 b(it)f(su\016ces)h (to)f(sho)m(w)g(that)773 4438 y(lim)17 b(sup)855 4518 y Fw(\017)p Fl(!)p Fv(0)1088 4438 y Fo(j)p Fp(\017)p Fo(j)1200 4303 y Fm(Z)1299 4329 y Fw(b)1255 4528 y(a)1350 4438 y Fo(k)p Fh(1)p 1456 4410 V 26 x Fw(R)1514 4438 y Fo(R)p Fr(\()p Fp(e)22 b Fr(+)g(i)p Fp(\017)p Fr(\))p Fh(1)1961 4453 y Fv(\002)2021 4438 y Fr(\()p Fp(H)8 b Fr(\))p Fp(\016)2229 4454 y Fv(\()p Fw(n;)p Fv(0\))2385 4438 y Fo(k)2435 4397 y Fv(2)2474 4438 y Fr(d)p Fp(e)29 b Fo(\024)f Fp(C)7 b Fr(e)2827 4397 y Fl(\000)p Fw(\015)t(R)2980 4438 y Fp(:)0 4705 y Fr(Since)34 b Fp(V)49 b Fr(=)27 b Fh(1)520 4720 y Fv(0)560 4705 y Fp(V)54 b Fr(and)993 4984 y(sup)1007 5066 y Fw(\017)p Fl(6)p Fv(=0)1157 4984 y Fo(j)p Fp(\017)p Fo(j)1269 4848 y Fm(Z)1368 4874 y Fv(b)1324 5074 y Fw(a)1428 4984 y Fo(k)p Fp(V)21 b Fo(R)p Fr(\()p Fp(e)i Fr(+)f(i)p Fp(\017)p Fr(\))p Fh(1)2004 4999 y Fv(\002)2063 4984 y Fr(\()p Fp(H)8 b Fr(\))p Fp(\016)2271 4999 y Fv(\()p Fw(n;)p Fv(0\))2428 4984 y Fo(k)2478 4943 y Fv(2)2545 4984 y Fo(\024)28 b Fp(C)2720 4999 y Fv(2)2759 4984 y Fp(;)0 5266 y Fr(the)33 b(result)h(follo)m(ws)f(from)f(the)h (resolv)m(en)m(t)i(iden)m(tit)m(y)f(and)f(Lemma)g(2.2.)43 b Fb(2)p eop %%Page: 9 9 9 8 bop 3731 100 a Fr(9)0 407 y(W)-8 b(e)33 b(are)g(no)m(w)g(ready)g (to)f(\014nish:)0 697 y Fh(Pro)s(of)42 b(of)g(Theorem)h(1.1.)55 b Fr(Let)37 b(\002)f(b)s(e)h(a)f(compact)h(set)h(suc)m(h)g(that)e(\002) 25 b Fo(\\)g Fp(\033)t Fr(\()p Fp(H)3028 712 y Fv(0)3067 697 y Fr(\))35 b(=)f Fo(;)p Fr(.)55 b(It)37 b(follo)m(ws)0 818 y(from)32 b(Lemmas)i(2.1)e(and)h(2.3)f(that)g(for)g(an)m(y)i Fp(n)28 b Fo(2)g Fn(Z)1912 782 y Fw(d)1953 818 y Fr(,)793 1100 y(lim)17 b(sup)847 1181 y Fw(T)10 b Fl(!1)1154 1032 y Fr(1)p 1119 1077 120 4 v 1119 1168 a(2)p Fp(T)1265 964 y Fm(Z)1364 990 y Fw(T)1320 1190 y Fl(\000)p Fw(T)1447 1100 y Fo(k)p Fh(1)p 1553 1071 58 3 v 26 x Fw(R)1610 1100 y Fr(e)1653 1059 y Fl(\000)p Fv(i)p Fw(tH)1821 1100 y Fh(1)1877 1115 y Fv(\002)1936 1100 y Fr(\()p Fp(H)e Fr(\))p Fp(\016)2144 1115 y Fv(\()p Fw(n;)p Fv(0\))2300 1100 y Fo(k)2350 1059 y Fv(2)2389 1100 y Fr(d)p Fp(t)28 b Fr(=)g Fp(O)s Fr(\(e)2769 1059 y Fl(\000)p Fw(\015)t(R)2921 1100 y Fr(\))p Fp(:)0 1387 y Fr(Th)m(us,)41 b(for)d(all)g Fp(n)f Fo(2)g Fn(Z)841 1351 y Fv(d)885 1387 y Fr(,)j Fh(1)1008 1402 y Fv(\002)1067 1387 y Fr(\()p Fp(H)8 b Fr(\))p Fp(\016)1275 1402 y Fv(\()p Fw(n;)p Fv(0\))1468 1387 y Fo(\032)37 b(H)1666 1402 y Fv(s)1699 1387 y Fr(\()p Fp(H)8 b Fr(\).)59 b(Since)39 b(the)g(set)f Fo(f)p Fp(\016)2634 1402 y Fv(\()p Fw(n;)p Fv(0\))2828 1387 y Fr(:)f Fp(n)h Fo(2)f Fn(Z)3157 1351 y Fw(d)3198 1387 y Fo(g)h Fr(is)g(cyclic)i(for)0 1507 y Fp(H)47 b Fr(\(see)41 b([JL1]\),)h(w)m(e)f(ha)m(v)m(e)g(that)f (Ran)p Fh(1)1485 1523 y Ff(R)p Fl(n)p Fv([)p Fl(\000)p Fw(c)1674 1535 y Fe(d)1709 1523 y Fw(;c)1760 1535 y Fe(d)1795 1523 y Fv(])1819 1507 y Fr(\()p Fp(H)8 b Fr(\))40 b Fo(\032)g(H)2225 1522 y Fv(s)2257 1507 y Fr(\()p Fp(H)8 b Fr(\).)65 b(Finally)-8 b(,)42 b(either)f Fh(1)3214 1523 y Fl(f\006)p Fw(c)3335 1535 y Fe(d)3371 1523 y Fl(g)3410 1507 y Fr(\()p Fp(H)8 b Fr(\))40 b(=)g(0)0 1628 y(or)e Fo(\006)p Fp(c)244 1643 y Fw(d)323 1628 y Fr(is)g(an)g(eigen)m(v)-5 b(alue)40 b(of)d Fp(H)8 b Fr(.)60 b(In)38 b(either)h(case,)i(Ran)o Fh(1)2211 1643 y Fl(f\006)p Fw(c)2332 1655 y Fe(d)2368 1643 y Fl(g)2408 1628 y Fr(\()p Fp(H)8 b Fr(\))36 b Fo(\032)i(H)2808 1643 y Fv(s)2840 1628 y Fr(\()p Fp(H)8 b Fr(\))37 b(and)h(Theorem)i (1.1)0 1748 y(follo)m(ws.)k Fb(2)0 2136 y Fi(2.2)135 b(Ranges)46 b(of)f FD(\012)1011 2093 y FB(\006)0 2371 y Fr(In)33 b(this)g(section)h(w)m(e)f(pro)m(v)m(e)h(Theorem)g(1.2.)0 2541 y(Let)865 2761 y Fo(D)d Fr(:=)1103 2680 y Fm(\010)1161 2761 y Fp( )h Fr(:)c Fo(8)p Fp(R)q(;)17 b(k)31 b Fo(\025)d Fr(0)p Fp(;)49 b Fo(k)p Fh(1)1903 2776 y Fw(R)1961 2761 y Fr(e)2004 2720 y Fl(\000)p Fv(i)p Fw(tH)2162 2729 y Fj(0)2201 2761 y Fp( )t Fo(k)28 b Fr(=)f Fp(O)s Fr(\()p Fo(h)p Fp(t)p Fo(i)2678 2720 y Fl(\000)p Fw(k)2775 2761 y Fr(\))2813 2680 y Fm(\011)2887 2761 y Fp(:)0 2981 y Fr(Ob)m(viously)-8 b(,)43 b Fo(D)e Fr(is)f(a)f(linear)g(subspace)i(of)e Fo(H)q Fr(.)63 b(Moreo)m(v)m(er,)42 b(in)d([JL1])g(\(Prop)s(osition)h (3.12\))e(w)m(e)i(ha)m(v)m(e)0 3102 y(sho)m(wn)34 b(that)p 1728 3242 80 4 v 1728 3322 a Fo(D)d Fr(=)c Fo(H)q Fp(:)1528 b Fr(\(2.9\))0 3591 y(Theorem)34 b(1.2)e(follo)m(ws)h(from)g(the)g (next)g(three)g(prop)s(ositions.)0 3811 y Fh(Prop)s(osition)38 b(2.4)49 b Fr(Ran)o(\012)1042 3775 y Fl(\006)1129 3811 y Fo(\032)p 1235 3731 V 29 w(T)25 b Fg(.)0 4048 y Fh(Pro)s(of.)65 b Fr(W)-8 b(e)33 b(consider)h(the)f Fo(\000)g Fr(case;)g(a)g(similar)g (argumen)m(t)g(applies)h(to)e(the)h(+)g(case.)0 4218 y(Let)g Fp(T)46 b Fr(b)s(e)33 b(a)f(linear)h(op)s(erator)f(de\014ned)i (b)m(y)1231 4551 y Fp(T)14 b(\016)1345 4567 y Fv(\()p Fw(n;x)p Fv(\))1534 4551 y Fr(=)1638 4347 y Fm(8)1638 4436 y(<)1638 4616 y(:)1768 4430 y Fo(\000)p Fp(\016)1888 4445 y Fv(\()p Fw(n;)p Fv(1\))2128 4430 y Fr(if)33 b Fp(x)28 b Fr(=)f(0)1806 4550 y Fp(\016)1849 4566 y Fv(\()p Fw(n;)p Fv(0\))2128 4550 y Fr(if)33 b Fp(x)28 b Fr(=)f(1)1882 4671 y(0)197 b(if)33 b Fp(x)28 b(>)f Fr(1.)3530 4551 y(\(2.10\))0 4889 y(Note)33 b(that)f Fo(k)p Fp(T)14 b Fo(k)27 b Fr(=)h(1)k(and)1295 5109 y Fp(H)8 b Fh(1)p 1440 5081 40 3 v 24 x Fv(0)1501 5109 y Fo(\000)23 b Fh(1)p 1657 5081 V 24 x Fv(0)1696 5109 y Fp(H)1777 5124 y Fv(0)1844 5109 y Fr(=)k([)p Fp(H)2055 5124 y Fv(0)2095 5109 y Fp(;)17 b Fh(1)p 2195 5081 V 24 x Fv(0)2234 5109 y Fr(])28 b(=)f Fp(T)8 b(:)1046 b Fr(\(2.11\))p eop %%Page: 10 10 10 9 bop 3682 100 a Fr(10)0 407 y(Fix)33 b Fp( )e Fo(2)e(D)s Fr(.)43 b(Since)1431 627 y(lim)1416 687 y Fw(t)p Fl(!1)1599 627 y Fr(e)1642 586 y Fv(i)p Fw(tH)1754 627 y Fh(1)1810 642 y Fv(0)1850 627 y Fr(e)1893 586 y Fl(\000)p Fv(i)p Fw(tH)2051 595 y Fj(0)2090 627 y Fp( )32 b Fr(=)27 b(0)p Fp(;)0 874 y Fr(w)m(e)34 b(ha)m(v)m(e)1126 1120 y(\012)1196 1078 y Fl(\000)1255 1120 y Fp( )26 b Fo(\000)d Fp( )32 b Fr(=)1642 984 y Fm(Z)1742 1010 y Fl(1)1698 1209 y Fv(0)1870 1052 y Fr(d)p 1843 1097 108 4 v 1843 1188 a(d)p Fp(\034)1961 1120 y Fr(e)2004 1078 y Fv(i)p Fw(\034)8 b(H)2130 1120 y Fh(1)p 2186 1091 40 3 v 23 x Fv(0)2225 1120 y Fr(e)2268 1078 y Fl(\000)p Fv(i)p Fw(\034)g(H)2440 1087 y Fj(0)2479 1120 y Fp( )t Fr(d)p Fp(\034)1539 1497 y Fr(=)27 b(i)1686 1362 y Fm(Z)1786 1388 y Fl(1)1741 1587 y Fv(0)1877 1497 y Fr(e)1920 1456 y Fv(i)p Fw(\034)8 b(H)2046 1497 y Fp(T)14 b Fr(e)2160 1456 y Fl(\000)p Fv(i)p Fw(\034)8 b(H)2332 1465 y Fj(0)2370 1497 y Fp( )t Fr(d)p Fp(\034)e(:)0 1764 y Fr(Th)m(us,)1023 1978 y Fo(k)p Fh(1)1129 1993 y Fw(R)1187 1978 y Fr(e)1230 1937 y Fl(\000)p Fv(i)p Fw(tH)1397 1978 y Fr(\012)1467 1937 y Fl(\000)1527 1978 y Fp( )t Fo(k)1644 1937 y Fv(2)1711 1978 y Fr(=)27 b Fo(k)p Fh(1)1920 1993 y Fw(R)1978 1978 y Fr(\012)2048 1937 y Fl(\000)2108 1978 y Fr(e)2151 1937 y Fl(\000)p Fv(i)p Fw(tH)2309 1946 y Fj(0)2348 1978 y Fp( )t Fo(k)2465 1937 y Fv(2)1711 2205 y Fo(\024)h Fp(A)p Fr(\()p Fp(t)p Fr(\))22 b(+)g(2)p Fo(k)p Fh(1)2275 2220 y Fw(R)2333 2205 y Fr(e)2376 2164 y Fl(\000)p Fv(i)p Fw(tH)2534 2173 y Fj(0)2573 2205 y Fp( )t Fo(k)2690 2164 y Fv(2)2729 2205 y Fp(;)3530 2091 y Fr(\(2.12\))0 2418 y(where)1122 2687 y Fp(A)p Fr(\()p Fp(t)p Fr(\))28 b(=)f(2)1503 2547 y Fm(\022)1576 2552 y(Z)1676 2578 y Fl(1)1632 2777 y Fv(0)1767 2687 y Fo(k)p Fp(T)14 b Fr(e)1931 2646 y Fl(\000)p Fv(i\()p Fw(\034)8 b Fv(+)p Fw(t)p Fv(\))p Fw(H)2237 2655 y Fj(0)2277 2687 y Fp( )t Fo(k)p Fr(d)p Fp(\034)2501 2547 y Fm(\023)2574 2569 y Fv(2)2630 2687 y Fp(:)0 2964 y Fr(Since)35 b Fp(T)43 b Fr(=)29 b Fp(T)14 b Fh(1)588 2979 y Fv(1)627 2964 y Fr(,)34 b(it)f(follo)m(ws)h(from)f(the)h(de\014nition)h(of)e Fo(D)j Fr(and)d(\(2.12\))g(that)g(for)g(an)m(y)i Fp( )e Fo(2)c(D)s Fr(,)34 b Fp(R)c Fo(\025)g Fr(0)0 3085 y(and)j Fp(k)d Fo(\025)f Fr(0,)1299 3305 y Fo(k)p Fh(1)1405 3320 y Fw(R)1463 3305 y Fr(e)1506 3264 y Fl(\000)p Fv(i)p Fw(tH)1673 3305 y Fr(\012)1743 3264 y Fl(\000)1803 3305 y Fp( )t Fo(k)1920 3264 y Fv(2)1987 3305 y Fr(=)e Fp(O)s Fr(\()p Fo(h)p Fp(t)p Fo(i)2319 3264 y Fl(\000)p Fw(k)2416 3305 y Fr(\))p Fp(:)0 3525 y Fr(Th)m(us,)34 b(\012)344 3489 y Fl(\000)404 3525 y Fo(D)c(\032)e(T)e Fr(,)32 b(and)h(relation)g (\(2.9\))f(yields)i(the)f(result.)44 b Fb(2)0 3795 y Fh(Prop)s(osition)38 b(2.5)p 798 3715 82 4 v 49 w Fo(F)f(\032)28 b Fr(Ran\012)1257 3758 y Fl(\006)1316 3795 y Fg(.)0 4031 y Fh(Pro)s(of.)64 b Fr(W)-8 b(e)32 b(again)f(consider)i(only)g(the)f Fo(\000)g Fr(case.)44 b(Ob)m(viously)-8 b(,)34 b(to)e(pro)m(v)m(e)h (the)f(statemen)m(t)i(it)d(su\016ces)0 4152 y(to)h(sho)m(w)i(that)e (for)g(ev)m(ery)j Fp( )c Fo(2)e(F)42 b Fr(the)33 b(limit)1582 4372 y(lim)1567 4431 y Fw(t)p Fl(!1)1750 4372 y Fr(e)1793 4330 y Fv(i)p Fw(tH)1896 4339 y Fj(0)1935 4372 y Fr(e)1978 4330 y Fl(\000)p Fv(i)p Fw(tH)2146 4372 y Fp( )1321 b Fr(\(2.13\))0 4615 y(exists.)0 4786 y(First,)33 b(w)m(e)g(observ)m(e)i (that)d(for)g Fp( )g Fo(2)c(F)10 b Fr(,)1446 5006 y(lim)1430 5065 y Fw(t)p Fl(!1)1613 5006 y Fr(e)1656 4964 y Fv(i)p Fw(tH)1769 5006 y Fh(1)1825 5021 y Fv(0)1865 5006 y Fr(e)1908 4964 y Fl(\000)p Fv(i)p Fw(tH)2075 5006 y Fp( )32 b Fr(=)27 b(0)p Fp(:)1181 b Fr(\(2.14\))p eop %%Page: 11 11 11 10 bop 3682 100 a Fr(11)0 407 y(Indeed,)43 b(let)c Fp(w)s Fr(\()p Fp(t)p Fr(\))g(=)g(e)881 371 y Fv(i)p Fw(tH)994 407 y Fh(1)1050 422 y Fv(0)1089 407 y Fr(e)1132 371 y Fl(\000)p Fv(i)p Fw(tH)1300 407 y Fp( )t Fr(.)64 b(Then)1720 327 y Fm(R)1803 407 y Fo(k)p Fp(w)s Fr(\()p Fp(t)p Fr(\))p Fo(k)2087 371 y Fv(2)2125 407 y Fr(d)p Fp(t)40 b(<)f Fo(1)p Fr(,)i(and,)g(since)g([)p Fp(H)r(;)17 b Fh(1)3218 422 y Fv(0)3257 407 y Fr(])40 b(=)f([)p Fp(H)3547 422 y Fv(0)3586 407 y Fp(;)17 b Fh(1)3686 422 y Fv(0)3725 407 y Fr(],)0 527 y Fo(k)p Fp(w)123 491 y Fl(0)146 527 y Fr(\()p Fp(t)p Fr(\))p Fo(k)27 b(\024)h Fr(2)p Fo(k)p Fp(H)619 542 y Fv(0)658 527 y Fo(k)p Fr(.)42 b(This)31 b(yields)h(that)d(lim)1612 542 y Fw(t)p Fl(!1)1800 527 y Fp(w)s Fr(\()p Fp(t)p Fr(\))e(=)h(0)h(\(see,)i(e.g.,)g(Exercise)h (6.2)d(in)h([RS]\).)g(Th)m(us,)0 648 y(to)i(pro)m(v)m(e)i(the)f (existence)i(of)d(the)h(limit)g(\(2.13\))f(it)h(su\016ces)h(to)f(pro)m (v)m(e)g(the)g(existence)j(of)1521 868 y(lim)1506 927 y Fw(t)p Fl(!1)1689 868 y Fr(e)1732 826 y Fv(i)p Fw(tH)1835 835 y Fj(0)1874 868 y Fh(1)p 1930 839 40 3 v 23 x Fv(0)1969 868 y Fr(e)2012 826 y Fl(\000)p Fv(i)p Fw(tH)2180 868 y Fp( )t(:)0 1164 y Fr(W)-8 b(e)33 b(\014x)g Fp( )f Fo(2)c(F)42 b Fr(and)33 b(set)1442 1384 y Fp(`)p Fr(\()p Fp(t)p Fr(\))28 b(:=)f(e)1795 1343 y Fv(i)p Fw(tH)1898 1352 y Fj(0)1937 1384 y Fh(1)p 1993 1356 V 24 x Fv(0)2033 1384 y Fr(e)2076 1343 y Fl(\000)p Fv(i)p Fw(tH)2243 1384 y Fp( )t(:)0 1604 y Fr(Let)33 b Fp(\036)27 b Fo(2)h(H)34 b Fr(b)s(e)f(arbitrary)-8 b(.)43 b(Then)1223 1800 y(d)p 1205 1845 90 4 v 1205 1936 a(d)p Fp(t)1305 1868 y Fr(\()p Fp(\036)p Fo(j)p Fp(w)s Fr(\()p Fp(t)p Fr(\)\))26 b(=)i(i\(e)1889 1827 y Fl(\000)p Fv(i)p Fw(tH)2047 1836 y Fj(0)2086 1868 y Fp(\036)p Fo(j)p Fp(T)14 b Fr(e)2286 1827 y Fl(\000)p Fv(i)p Fw(tH)2452 1868 y Fp( )t Fr(\))p Fp(;)0 2119 y Fr(where)34 b Fp(T)46 b Fr(is)33 b(giv)m(en)h(b)m(y)f(\(2.10\).)43 b(Since)34 b Fp(T)41 b Fr(=)28 b Fp(T)14 b Fh(1)1777 2134 y Fv(1)1844 2119 y Fr(=)27 b Fh(1)2003 2134 y Fv(1)2043 2119 y Fp(T)14 b Fr(,)32 b(w)m(e)i(ha)m(v)m(e)f(that)g(for)f Fp(t)c(>)f(s)p Fr(,)487 2435 y Fo(j)p Fr(\()p Fp(\036)p Fo(j)p Fp(`)p Fr(\()p Fp(t)p Fr(\))21 b Fo(\000)i Fp(`)p Fr(\()p Fp(s)p Fr(\)\))p Fo(j)k(\024)1273 2294 y Fm(\022)1346 2299 y(Z)1402 2525 y Ff(R)1470 2435 y Fo(k)p Fh(1)1576 2450 y Fv(1)1616 2435 y Fr(e)1659 2394 y Fl(\000)p Fv(i)p Fw(\034)8 b(H)1831 2403 y Fj(0)1869 2435 y Fp(\036)p Fo(k)1977 2394 y Fv(2)2016 2435 y Fr(d)p Fp(\034)2123 2294 y Fm(\023)2207 2289 y Fj(1)p 2207 2301 31 3 v 2207 2343 a(2)2268 2294 y Fm(\022)2342 2299 y(Z)2441 2325 y Fw(t)2397 2525 y(s)2488 2435 y Fo(k)p Fh(1)2594 2450 y Fv(1)2633 2435 y Fr(e)2676 2394 y Fl(\000)p Fv(i)p Fw(\034)g(H)2857 2435 y Fp( )t Fo(k)2974 2394 y Fv(2)3013 2435 y Fr(d)p Fp(\034)3120 2294 y Fm(\023)3204 2284 y Fj(1)p 3204 2296 V 3204 2337 a(2)3265 2435 y Fp(:)0 2712 y Fr(Since)32 b Fh(1)309 2727 y Fv(1)379 2712 y Fr(is)f Fp(H)556 2727 y Fv(0)595 2712 y Fr(-smo)s(oth)f(\(see)h(Lemma)h (3.7)e(in)g([JL1]\),)h(there)h(is)f(a)f(constan)m(t)h Fp(C)7 b Fr(,)31 b(indep)s(enden)m(t)i(of)d Fp(\036)p Fr(,)0 2832 y(suc)m(h)k(that)1203 2977 y Fm(\022)1277 2982 y(Z)1332 3208 y Ff(R)1401 3118 y Fo(k)p Fh(1)1507 3133 y Fv(1)1546 3118 y Fr(e)1589 3076 y Fl(\000)p Fv(i)p Fw(\034)8 b(H)1761 3085 y Fj(0)1800 3118 y Fp(\036)p Fo(k)1908 3076 y Fv(2)1946 3118 y Fr(d)p Fp(\034)2053 2977 y Fm(\023)2137 2972 y Fj(1)p 2137 2984 V 2137 3026 a(2)2210 3118 y Fo(\024)28 b Fp(C)7 b Fo(k)p Fp(\036)p Fo(k)p Fp(:)0 3389 y Fr(Therefore,)996 3700 y Fo(k)p Fp(`)p Fr(\()p Fp(t)p Fr(\))22 b Fo(\000)h Fp(`)p Fr(\()p Fp(s)p Fr(\))p Fo(k)k(\024)h Fp(C)1759 3559 y Fm(\022)1832 3564 y(Z)1932 3590 y Fw(t)1888 3790 y(s)1978 3700 y Fo(k)p Fh(1)2084 3715 y Fv(1)2124 3700 y Fr(e)2167 3659 y Fl(\000)p Fv(i)p Fw(\034)8 b(H)2348 3700 y Fp( )t Fo(k)2465 3659 y Fv(2)2504 3700 y Fr(d)p Fp(\034)2611 3559 y Fm(\023)2695 3549 y Fj(1)p 2695 3561 V 2695 3602 a(2)2756 3700 y Fp(:)0 3981 y Fr(If)42 b Fp( )49 b Fo(2)44 b(F)10 b Fr(,)45 b(then)e(the)f(in)m(tegrand)h(on)g(the)f(righ)m(t)h(hand)g(side)g(of)f (the)h(last)f(equation)h(is)g(in)g Fp(L)3565 3945 y Fv(1)3605 3981 y Fr(\()p Fn(R)p Fr(\).)0 4102 y(Therefore,)34 b(the)f(sequence)i Fp(`)p Fr(\()p Fp(t)p Fr(\))e(is)g(Cauc)m(h)m(y)h(as)f Fp(t)28 b Fo(!)f(1)32 b Fr(and)h(lim)2441 4117 y Fw(t)p Fl(!1)2628 4102 y Fp(`)p Fr(\()p Fp(t)p Fr(\))f(exist.)45 b Fb(2)0 4371 y Fh(Prop)s(osition)38 b(2.6)p 798 4291 82 4 v 49 w Fo(F)f(?)28 b(H)1096 4386 y Fv(s)1128 4371 y Fr(\()p Fp(H)8 b Fr(\))p Fg(.)0 4608 y Fh(Pro)s(of.)65 b Fr(Let)33 b Fp(\036)27 b Fo(2)h(F)42 b Fr(and)33 b Fp( )f Fo(2)c(H)1299 4623 y Fv(s)1331 4608 y Fr(\()p Fp(H)8 b Fr(\))32 b(b)s(e)g(t)m(w)m(o)i(normalized)f(v)m(ectors.)45 b(Then)759 4828 y(\()p Fp(\036)p Fo(j)p Fp( )t Fr(\))27 b(=)h(\()p Fh(1)1213 4843 y Fw(R)1271 4828 y Fr(e)1314 4787 y Fl(\000)p Fv(i)p Fw(tH)1481 4828 y Fp(\036)p Fo(j)p Fh(1)1623 4843 y Fw(R)1680 4828 y Fr(e)1723 4787 y Fl(\000)p Fv(i)p Fw(tH)1891 4828 y Fp( )t Fr(\))22 b(+)g(\()p Fh(1)p 2210 4800 58 3 v 26 x Fw(R)2268 4828 y Fr(e)2311 4787 y Fl(\000)p Fv(i)p Fw(tH)2478 4828 y Fp(\036)p Fo(j)p Fh(1)p 2620 4800 V 26 x Fw(R)2677 4828 y Fr(e)2720 4787 y Fl(\000)p Fv(i)p Fw(tH)2888 4828 y Fp( )t Fr(\))p Fp(;)0 5048 y Fr(whic)m(h)34 b(yields)1122 5268 y Fo(j)p Fr(\()p Fp(\036)p Fo(j)p Fp( )t Fr(\))p Fo(j)26 b(\024)i(k)p Fh(1)1644 5283 y Fw(R)1702 5268 y Fr(e)1745 5227 y Fl(\000)p Fv(i)p Fw(tH)1912 5268 y Fp(\036)p Fo(k)22 b Fr(+)g Fo(k)p Fh(1)p 2246 5239 V 26 x Fw(R)2303 5268 y Fr(e)2346 5227 y Fl(\000)p Fv(i)p Fw(tH)2514 5268 y Fp( )t Fo(k)p Fp(:)p eop %%Page: 12 12 12 11 bop 3682 100 a Fr(12)0 407 y(Th)m(us,)445 688 y Fo(j)p Fr(\()p Fp(\036)p Fo(j)p Fp( )t Fr(\))p Fo(j)27 b(\024)862 548 y Fm(\022)980 621 y Fr(1)p 945 665 120 4 v 945 757 a(2)p Fp(T)1091 553 y Fm(Z)1191 579 y Fw(T)1147 778 y Fl(\000)p Fw(T)1273 688 y Fo(k)p Fh(1)1379 703 y Fw(R)1437 688 y Fr(e)1480 647 y Fl(\000)p Fv(i)p Fw(tH)1647 688 y Fp(\036)p Fo(k)1755 647 y Fv(2)1794 688 y Fr(d)p Fp(t)1883 548 y Fm(\023)1967 533 y Fj(1)p 1967 545 31 3 v 1967 586 a(2)2034 688 y Fr(+)2132 548 y Fm(\022)2250 621 y Fr(1)p 2215 665 120 4 v 2215 757 a(2)p Fp(T)2361 553 y Fm(Z)2461 579 y Fw(T)2416 778 y Fl(\000)p Fw(T)2543 688 y Fo(k)p Fh(1)p 2649 660 58 3 v 26 x Fw(R)2706 688 y Fr(e)2749 647 y Fl(\000)p Fv(i)p Fw(tH)2917 688 y Fp( )t Fo(k)3034 647 y Fv(2)3073 688 y Fr(d)p Fp(t)3162 548 y Fm(\023)3246 533 y Fj(1)p 3246 545 31 3 v 3246 586 a(2)3307 688 y Fp(;)0 929 y Fr(and)645 1191 y Fo(j)p Fr(\()p Fp(\036)p Fo(j)p Fp( )t Fr(\))p Fo(j)f(\024)83 b Fr(lim)17 b(sup)1170 1272 y Fw(T)10 b Fl(!1)1432 1051 y Fm(\022)1551 1124 y Fr(1)p 1515 1168 120 4 v 1515 1259 a(2)p Fp(T)1661 1055 y Fm(Z)1761 1082 y Fw(T)1717 1281 y Fl(\000)p Fw(T)1843 1191 y Fo(k)p Fh(1)1949 1206 y Fw(R)2007 1191 y Fr(e)2050 1150 y Fl(\000)p Fv(i)p Fw(tH)2217 1191 y Fp(\036)p Fo(k)2325 1150 y Fv(2)2364 1191 y Fr(d)p Fp(t)2453 1051 y Fm(\023)2537 1035 y Fj(1)p 2537 1047 31 3 v 2537 1089 a(2)1116 1533 y Fr(+)46 b(lim)1209 1595 y Fw(R)p Fl(!1)1420 1533 y Fr(lim)17 b(sup)1474 1614 y Fw(T)10 b Fl(!1)1736 1392 y Fm(\022)1854 1466 y Fr(1)p 1819 1510 120 4 v 1819 1601 a(2)p Fp(T)1965 1397 y Fm(Z)2065 1424 y Fw(T)2020 1623 y Fl(\000)p Fw(T)2147 1533 y Fo(k)p Fh(1)p 2253 1505 58 3 v 26 x Fw(R)2311 1533 y Fr(e)2354 1492 y Fl(\000)p Fv(i)p Fw(tH)2521 1533 y Fp( )t Fo(k)2638 1492 y Fv(2)2677 1533 y Fr(d)p Fp(t)2766 1392 y Fm(\023)2850 1377 y Fj(1)p 2850 1389 31 3 v 2850 1431 a(2)2939 1533 y Fr(=)44 b(0)p Fp(:)0 1749 y Fb(2)0 2157 y Fq(3)161 b(Surface)53 b(sp)t(ectra)g(in)h(dimension)e(t)l(w)l(o)0 2416 y Fr(In)30 b(this)f(section)i(w)m(e)f(pro)m(v)m(e)g(Theorems)h (1.4)e(and)h(1.5.)42 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Fv(2)2649 4452 y Fo(\000)g Fr(4)2798 4482 y Fl(\006)2857 4342 y Fm(\021)3530 4452 y Fr(\(3.15\))0 4664 y(where)34 b(the)f(branc)m(h)g(of)f(square)i(ro)s(ot)e(is)h(\014xed)h(b)m(y)459 4839 y Fo(p)p 542 4839 73 4 v 77 x Fp(w)614 4939 y Fl(\006)701 4916 y Fr(=)804 4828 y Fm(p)p 904 4828 254 4 v 88 x Fp(x)23 b Fr(+)f(i)p Fp(y)1158 4947 y Fl(\006)1245 4916 y Fr(=)1358 4766 y Fo(p)p 1441 4766 49 4 v 83 x Fr(2)p 1358 4893 132 4 v 1400 4984 a(2)1516 4805 y Fm(\020)1576 4826 y(p)p 1676 4826 304 4 v 90 x Fo(j)p Fp(w)s Fo(j)f Fr(+)h Fp(x)g Fo(\006)h Fr(i)2128 4826 y Fm(p)p 2227 4826 305 4 v 2227 4916 a Fo(j)p Fp(w)s Fo(j)e(\000)i Fp(x)2532 4805 y Fm(\021)2608 4916 y Fp(;)212 b Fo(\006)p Fr(Im)q Fp(w)30 b(>)e Fr(0)p Fp(:)0 5147 y Fr(The)i(function)g Fp(\025)p Fr(\()p Fp(\036;)17 b(z)t Fr(\))29 b(extends)i(b)m(y)f(con)m(tin)m(uit)m(y)h(from)e Fn(T)15 b Fo(\002)g Fn(C)2302 5162 y Fl(\006)2391 5147 y Fr(to)29 b Fn(T)15 b Fo(\002)p 2680 5067 72 4 v 15 w Fn(C)2753 5162 y Fl(\006)2812 5147 y Fr(.)43 b(W)-8 b(e)29 b(denote)h(the)g(v)-5 b(alues)0 5268 y(of)32 b(these)i (extensions)h(along)d(the)h(real)g(axis)g(b)m(y)g Fp(\025)p Fr(\()p Fp(\036;)17 b(e)22 b Fo(\006)h Fr(i0\).)p eop %%Page: 13 13 13 12 bop 3682 100 a Fr(13)0 407 y Fh(Lemma)39 b(3.1)49 b Fg(F)-7 b(or)34 b(any)h Fp(x)28 b Fo(2)g Fn(Z)1201 422 y Fv(+)1295 407 y Fg(ther)-5 b(e)35 b(is)g(a)f(c)-5 b(onstant)35 b Fp(C)2186 422 y Fw(x)2265 407 y Fg(such)f(that)h Fo(8)p Fp(n)p Fg(,)1104 681 y Fr(sup)1086 763 y Fw(z)s Fl(2)p Ff(C)1217 772 y Fk(\006)1286 537 y Fm(\014)1286 596 y(\014)1286 656 y(\014)1286 716 y(\014)1319 545 y(Z)1374 771 y Ff(T)1439 681 y Fr(e)1482 640 y Fl(\000)p Fv(i)p Fw(n\036)1646 681 y Fp(\025)p Fr(\()p Fp(\036;)17 b(z)t Fr(\))1930 640 y Fw(x)1974 681 y Fr(d)p Fp(\036)2086 537 y Fm(\014)2086 596 y(\014)2086 656 y(\014)2086 716 y(\014)2147 681 y Fo(\024)28 b Fp(C)2322 696 y Fw(x)2366 681 y Fo(h)p Fp(n)p Fo(i)2502 640 y Fl(\000)p Fv(3)p Fw(=)p Fv(2)2666 681 y Fp(:)837 b Fr(\(3.16\))0 1138 y Fh(Pro)s(of.)65 b Fr(In)33 b([JL1])f(w)m(e)i(ha)m(v)m(e)g(sho)m(wn)g (that)680 1406 y(\()p Fp(\016)761 1422 y Fv(\(0)p Fw(;)p Fv(0\))910 1406 y Fo(j)p Fr(\()p Fp(H)1057 1421 y Fv(0)1118 1406 y Fo(\000)23 b Fp(z)t Fr(\))1305 1365 y Fl(\000)p Fv(1)1400 1406 y Fp(\016)1443 1422 y Fv(\()p Fw(n;x)p Fv(\))1604 1406 y Fr(\))28 b(=)f Fo(\000)p Fr(\(2)p Fp(\031)t Fr(\))2034 1365 y Fl(\000)p Fv(1)p Fw(=)p Fv(2)2216 1271 y Fm(Z)2271 1496 y Ff(T)2336 1406 y Fr(e)2379 1365 y Fl(\000)p Fv(i)p Fw(n\036)2543 1406 y Fp(\025)p Fr(\()p Fp(\036;)17 b(z)t Fr(\))2827 1365 y Fw(x)p Fv(+1)2961 1406 y Fr(d)p Fp(\036:)430 b Fr(\(3.17\))0 1679 y(Moreo)m(v)m(er,)34 b(the)f(pro)s(of)f(of)g(Lemma)h(2.9)f(in)h([JM2)q(])f(yields)i(that)f (for)f(some)h Fp(D)2843 1694 y Fw(x)2920 1679 y Fr(and)f(all)h Fp(n)p Fr(,)1128 1956 y(sup)1120 2042 y Fl(j)p Fw(e)p Fl(j)p Fw(>)p Fv(4)1316 1812 y Fm(\014)1316 1871 y(\014)1316 1931 y(\014)1316 1991 y(\014)1349 1820 y(Z)1404 2046 y Ff(T)1469 1956 y Fr(e)1512 1915 y Fl(\000)p Fv(i)p Fw(n\036)1676 1956 y Fp(\025)p Fr(\()p Fp(\036;)17 b(e)p Fr(\))1956 1915 y Fw(x)1999 1956 y Fr(d)p Fp(\036)2111 1812 y Fm(\014)2111 1871 y(\014)2111 1931 y(\014)2111 1991 y(\014)2172 1956 y Fo(\024)28 b Fp(D)2358 1971 y Fw(x)2402 1956 y Fo(h)p Fp(n)p Fo(i)2538 1915 y Fl(\000)p Fv(2)2632 1956 y Fp(:)0 2245 y Fr(Th)m(us,)50 b(the)45 b(maxim)m(um)h(mo)s(dulus)g(principle)g(and)f(the)h(standard)f(exp)s (onen)m(tial)h(estimate)g(on)f(the)0 2366 y(Green)33 b(function)g(\(3.17\))f(yield)i(that)e(\(3.16\))g(follo)m(ws)h(if)f (for)g(some)i Fp(C)2554 2381 y Fw(x)2630 2366 y Fr(and)f(all)f Fp(n)999 2643 y Fr(sup)992 2729 y Fl(j)p Fw(e)p Fl(j\024)p Fv(4)1187 2499 y Fm(\014)1187 2558 y(\014)1187 2618 y(\014)1187 2678 y(\014)1220 2508 y(Z)1276 2733 y Ff(T)1341 2643 y Fr(e)1384 2602 y Fl(\000)p Fv(i)p Fw(n\036)1547 2643 y Fp(\025)p Fr(\()p Fp(\036;)17 b(e)22 b Fo(\006)h Fr(i0\))2025 2602 y Fw(x)2068 2643 y Fr(d)p Fp(\036)2180 2499 y Fm(\014)2180 2558 y(\014)2180 2618 y(\014)2180 2678 y(\014)2241 2643 y Fo(\024)28 b Fp(C)2416 2658 y Fw(x)2460 2643 y Fo(h)p Fp(n)p Fo(i)2596 2602 y Fl(\000)p Fv(3)p Fw(=)p Fv(2)2761 2643 y Fp(:)0 2933 y Fr(The)42 b(pro)s(of)d(of)h(this)i(fact)e(is)h(a)g (calculus)h(exercise.)70 b(W)-8 b(e)41 b(indicate)g(b)s(elo)m(w)h(the)f (main)g(steps)h(of)e(this)0 3053 y(calculation.)0 3223 y(F)-8 b(or)43 b(notational)h(simplicit)m(y)-8 b(,)50 b(w)m(e)45 b(consider)g(the)g(cases)h Fp(x)i Fr(=)f(1,)g(+i0,)g(and)d Fp(e)k Fo(\025)g Fr(0.)78 b(W)-8 b(e)45 b(need)g(to)0 3344 y(estimate)34 b(the)f(in)m(tegral)1119 3606 y Fp(I)1162 3621 y Fw(n)1236 3606 y Fr(=)1340 3471 y Fm(Z)1395 3696 y Ff(T)1460 3606 y Fr(e)1503 3565 y Fl(\000)p Fv(i)p Fw(n\036)1667 3516 y Fm(p)p 1767 3516 724 4 v 90 x Fr(\(2)17 b(cos)f Fp(\036)22 b Fo(\000)h Fp(e)p Fr(\))2280 3578 y Fv(2)2341 3606 y Fo(\000)g Fr(4)2490 3636 y Fv(+)2549 3606 y Fr(d)p Fp(\036)0 3879 y Fr(for)29 b Fp(e)f Fo(2)g Fr([0)p Fp(;)17 b Fr(4])28 b(and)i Fo(j)p Fp(n)p Fo(j)d(\035)g Fr(1.)43 b(The)30 b(c)m(hange)g(of)f(v)-5 b(ariable)29 b Fp(u)f Fr(=)f(cos)17 b Fp(\036)29 b Fr(and)h(in)m(tegration)f(b)m(y)i (parts)e(yield)0 3999 y(that)1379 4219 y Fo(j)p Fp(I)1450 4234 y Fw(n)1496 4219 y Fo(j)f(\024)g Fr(\()p Fp(A)1768 4234 y Fw(n)p Fv(+)1892 4219 y Fr(+)22 b Fp(A)2063 4234 y Fw(n)p Fl(\000)2165 4219 y Fr(\))2203 4139 y Fm(\016)2260 4219 y Fo(j)p Fp(n)p Fo(j)p Fp(;)0 4439 y Fr(where)973 4722 y Fp(A)1046 4737 y Fw(n)p Fl(\006)1175 4722 y Fr(=)1279 4547 y Fm(\014)1279 4607 y(\014)1279 4667 y(\014)1279 4727 y(\014)1279 4787 y(\014)1312 4586 y(Z)1412 4612 y Fv(1)1367 4812 y Fl(\000)p Fv(1)1478 4722 y Fr(e)1521 4681 y Fl(\000)p Fv(i)p Fw(n)12 b Fv(cos)1745 4657 y Fk(\000)p Fj(1)1840 4681 y Fw(u)2025 4654 y Fr(2\(2)p Fp(u)21 b Fo(\006)i Fp(e)p Fr(\))p 1895 4699 657 4 v 1895 4719 a Fm(p)p 1994 4719 558 4 v 1994 4804 a Fr(\(2)p Fp(u)f Fo(\006)g Fp(e)p Fr(\))2341 4775 y Fv(2)2403 4804 y Fo(\000)g Fr(4)2561 4838 y Fv(+)2620 4722 y Fr(d)p Fp(u)2730 4547 y Fm(\014)2730 4607 y(\014)2730 4667 y(\014)2730 4727 y(\014)2730 4787 y(\014)2780 4722 y Fp(:)0 5023 y Fr(W)-8 b(e)33 b(estimate)h Fp(A)632 5038 y Fw(n)p Fv(+)734 5023 y Fr(.)43 b(Let)33 b(0)27 b Fp(<)h(\016)k Fo(\034)27 b Fr(1)32 b(and)931 5243 y Fo(S)7 b Fr(\()p Fp(\016)t Fr(\))28 b(=)g Fo(f)p Fp(u)f Fo(2)h Fr([)p Fo(\000)p Fr(1)p Fp(;)17 b Fr(1])44 b(:)g(dist)q(\()p Fp(u;)17 b Fo(\006)p Fr(1)22 b Fo(\000)g Fp(e=)p Fr(2\))28 b Fo(\024)g Fp(\016)t Fo(g)p Fp(:)p eop %%Page: 14 14 14 13 bop 3682 100 a Fr(14)0 407 y(Let)33 b(\011)251 422 y Fw(n)p Fv(+)385 407 y Fr(b)s(e)g(the)g(in)m(tegrand)g(in)g Fp(A)1309 422 y Fw(n)p Fv(+)1411 407 y Fr(.)43 b(One)33 b(can)g(easily)h(see)g(that)1382 527 y Fm(\014)1382 587 y(\014)1382 647 y(\014)1382 707 y(\014)1416 536 y(Z)1471 762 y Fl(S)5 b Fv(\()p Fw(\016)r Fv(\))1578 672 y Fr(\011)1654 687 y Fw(n)p Fv(+)1756 672 y Fr(\()p Fp(u)p Fr(\)d)p Fp(u)1998 527 y Fm(\014)1998 587 y(\014)1998 647 y(\014)1998 707 y(\014)2058 672 y Fo(\024)28 b Fp(C)2240 582 y Fo(p)p 2323 582 47 4 v 90 x Fp(\016)t(:)0 952 y Fr(If)33 b Fo(S)165 916 y Fv(c)201 952 y Fr(\()p Fp(\016)t Fr(\))27 b(=)h([)p Fo(\000)p Fr(1)p Fp(;)17 b Fr(1])22 b Fo(n)g(S)7 b Fr(\()p Fp(\016)t Fr(\),)33 b(then)289 1106 y Fm(\014)289 1165 y(\014)289 1225 y(\014)289 1285 y(\014)322 1114 y(Z)377 1340 y Fl(S)425 1321 y Fj(c)456 1340 y Fv(\()p Fw(\016)r Fv(\))516 1250 y Fr(\011)592 1265 y Fw(n)p Fv(+)693 1250 y Fr(\()p Fp(u)p Fr(\)d)p Fp(u)935 1106 y Fm(\014)935 1165 y(\014)935 1225 y(\014)935 1285 y(\014)995 1250 y Fr(=)1141 1183 y(1)p 1109 1227 114 4 v 1109 1318 a Fo(j)p Fp(n)p Fo(j)1249 1076 y Fm(\014)1249 1136 y(\014)1249 1195 y(\014)1249 1255 y(\014)1249 1315 y(\014)1282 1114 y(Z)1338 1340 y Fl(S)1386 1321 y Fj(c)1417 1340 y Fv(\()p Fw(\016)r Fv(\))1526 1110 y Fm(\022)1637 1183 y Fp(d)p 1609 1227 107 4 v 1609 1318 a(du)1725 1250 y Fr(e)1768 1209 y Fl(\000)p Fv(i)p Fw(n)12 b Fv(cos)1992 1186 y Fk(\000)p Fj(1)2087 1209 y Fw(u)2132 1110 y Fm(\023)2222 1163 y Fo(p)p 2305 1163 266 4 v 87 x Fr(1)22 b Fo(\000)g Fp(u)2531 1221 y Fv(2)2711 1183 y Fr(2\(2)p Fp(u)f Fr(+)h Fp(e)p Fr(\))p 2580 1227 656 4 v 2580 1247 a Fm(p)p 2680 1247 556 4 v 85 x Fr(\(2)p Fp(u)f Fr(+)h Fp(e)p Fr(\))3025 1303 y Fv(2)3087 1332 y Fo(\000)g Fr(4)3245 1367 y Fv(+)3304 1250 y Fr(d)p Fp(u)3414 1076 y Fm(\014)3414 1136 y(\014)3414 1195 y(\014)3414 1255 y(\014)3414 1315 y(\014)3464 1250 y Fp(;)0 1543 y Fr(and)33 b(in)m(tegration)g(b)m(y)g(parts)g(yields) 1339 1661 y Fm(\014)1339 1721 y(\014)1339 1780 y(\014)1339 1840 y(\014)1372 1670 y(Z)1427 1895 y Fl(S)1475 1876 y Fj(c)1506 1895 y Fv(\()p Fw(\016)r Fv(\))1566 1805 y Fr(\011)1642 1820 y Fw(n)p Fv(+)1743 1805 y Fr(\()p Fp(u)p Fr(\)d)p Fp(u)1985 1661 y Fm(\014)1985 1721 y(\014)1985 1780 y(\014)1985 1840 y(\014)2045 1805 y Fo(\024)2244 1738 y Fp(C)p 2160 1782 244 4 v 2160 1887 a Fo(j)p Fp(n)p Fo(j)2274 1802 y(p)p 2357 1802 47 4 v 85 x Fp(\016)2414 1805 y(:)0 2080 y Fr(Therefore,)h(for)e(some)h(constan)m(t)h Fp(C)39 b Fr(indep)s(enden)m(t)c(of)d Fp(e)p Fr(,)1324 2343 y Fo(j)p Fp(I)1395 2358 y Fw(n)1441 2343 y Fo(j)c(\024)1630 2275 y Fp(C)p 1612 2320 114 4 v 1612 2411 a Fo(j)p Fp(n)p Fo(j)1752 2202 y Fm(\022)1825 2253 y Fo(p)p 1908 2253 47 4 v 90 x Fp(\016)e Fr(+)2183 2275 y(1)p 2085 2320 244 4 v 2085 2425 a Fo(j)p Fp(n)p Fo(j)2199 2340 y(p)p 2282 2340 47 4 v 85 x Fp(\016)2339 2202 y Fm(\023)2429 2343 y Fp(:)0 2627 y Fr(Setting)33 b Fp(\016)f Fr(=)27 b Fo(j)p Fp(n)p Fo(j)628 2591 y Fl(\000)p Fv(1)722 2627 y Fr(,)33 b(w)m(e)g(deriv)m(e)i(the)e(statemen)m(t.)45 b Fb(2)0 2842 y Fr(Let)924 3077 y Fp(j)6 b Fr(\()p Fp(n;)17 b(z)t Fr(\))28 b(=)f(\(2)p Fp(\031)t Fr(\))1512 3035 y Fl(\000)p Fv(1)1623 2941 y Fm(Z)1678 3166 y Ff(T)1743 3077 y Fr(e)1786 3035 y Fl(\000)p Fv(i)p Fw(n\036)1950 3077 y Fr(\()p Fp(\025)p Fr(\()p Fp(\036;)17 b(z)t Fr(\))22 b(+)g(2)17 b(cos)g Fp(\036)p Fr(\)d)p Fp(\036;)690 b Fr(\(3.18\))0 3344 y(and)39 b(let)g Fp(h)399 3359 y Fv(0)438 3344 y Fr(\()p Fp(z)t Fr(\))h(b)s(e)e(the)h(op)s(erator)f(of)h(con)m(v) m(olution)h(b)m(y)f Fp(j)6 b Fr(\()p Fp(n;)17 b(z)t Fr(\).)62 b(This)40 b(op)s(erator)e(acts)h(on)g Fp(l)3472 3308 y Fv(2)3511 3344 y Fr(\()p Fn(Z)p Fr(\))g(as)0 3465 y(follo)m(ws:)1158 3675 y(\()p Fp(h)1252 3690 y Fv(0)1292 3675 y Fr(\()p Fp(z)t Fr(\))p Fp( )t Fr(\)\()p Fp(n)p Fr(\))28 b(=)1788 3581 y Fm(X)1840 3793 y Fw(k)1948 3675 y Fp(j)6 b Fr(\()p Fp(n)22 b Fo(\000)h Fp(k)s(;)17 b(z)t Fr(\))p Fp( )t Fr(\()p Fp(k)s Fr(\))p Fp(:)0 3977 y Fr(Let)31 b Fp(h)p Fr(\()p Fp(z)t Fr(\))d(=)g Fp(h)542 3992 y Fv(0)581 3977 y Fr(\()p Fp(z)t Fr(\))19 b(+)f Fp(V)52 b Fr(and)31 b Fp(H)k Fr(=)27 b Fp(H)1416 3992 y Fv(0)1474 3977 y Fr(+)18 b Fp(V)j Fr(,)31 b(where)h Fp(V)52 b Fr(is)31 b(a)g(b)s(oundary)g(p)s (oten)m(tial.)43 b(Note)30 b(that)h Fp(h)p Fr(\()p Fp(z)t Fr(\))0 4097 y(acts)i(on)g Fp(l)368 4061 y Fv(2)407 4097 y Fr(\()p Fn(Z)p Fr(\))g(while)h Fp(H)40 b Fr(acts)33 b(on)f Fp(l)1325 4061 y Fv(2)1365 4097 y Fr(\()p Fn(Z)1469 4061 y Fv(2)1469 4122 y(+)1528 4097 y Fr(\).)44 b(Let)742 4307 y(^)716 4332 y Fo(R)p Fr(\(\()p Fp(m;)17 b Fr(0\))p Fp(;)g Fr(\()p Fp(\036;)g(x)p Fr(\);)g Fp(z)t Fr(\))27 b(=)h(\(2)p Fp(\031)t Fr(\))1815 4291 y Fl(\000)1880 4264 y Fj(1)p 1879 4276 31 3 v 1879 4317 a(2)1940 4237 y Fm(X)1991 4446 y Fw(n)2101 4332 y Fo(R)p Fr(\(\()p Fp(m;)17 b Fr(0\))p Fp(;)g Fr(\()p Fp(n;)g(x)p Fr(\);)g Fp(z)t Fr(\)e)2928 4291 y Fv(i)p Fw(n\036)3037 4332 y Fp(:)0 4630 y Fr(F)-8 b(or)32 b(the)h(pro)s(of)e(of)i(the)g(next)g(t)m (w)m(o)g(lemmas)h(w)m(e)g(refer)f(the)g(reader)g(to)f([JL1,)h(JM1].)0 4846 y Fh(Lemma)39 b(3.2)49 b Fg(F)-7 b(or)34 b(any)h Fp(n;)17 b(m)28 b Fo(2)g Fn(Z)35 b Fg(and)f Fp(z)f Fo(2)28 b Fn(C)1801 4861 y Fl(\006)1860 4846 y Fg(,)920 5057 y Fr(\()p Fp(\016)1001 5072 y Fv(\()p Fw(m;)p Fv(0\))1178 5057 y Fo(j)p Fr(\()p Fp(H)h Fo(\000)23 b Fp(z)t Fr(\))1541 5016 y Fl(\000)p Fv(1)1636 5057 y Fp(\016)1679 5072 y Fv(\()p Fw(n;)p Fv(0\))1836 5057 y Fr(\))k(=)h(\()p Fp(\016)2086 5072 y Fw(m)2153 5057 y Fo(j)p Fr(\()p Fp(h)p Fr(\()p Fp(z)t Fr(\))22 b Fo(\000)h Fp(z)t Fr(\))2609 5016 y Fl(\000)p Fv(1)2704 5057 y Fp(\016)2747 5072 y Fw(n)2794 5057 y Fr(\))p Fp(:)p eop %%Page: 15 15 15 14 bop 3682 100 a Fr(15)0 407 y Fh(Lemma)39 b(3.3)49 b Fg(F)-7 b(or)34 b(any)h Fp(x)28 b Fo(2)g Fn(Z)1201 422 y Fv(+)1295 407 y Fg(and)35 b Fp(z)d Fo(2)c Fn(C)1728 422 y Fl(\006)1787 407 y Fg(,)892 575 y Fr(^)866 600 y Fo(R)q Fr(\(\()p Fp(m;)17 b Fr(0\))p Fp(;)g Fr(\()p Fp(\036;)g(x)p Fr(\))p Fp(;)g(z)t Fr(\))27 b(=)1807 575 y(^)1781 600 y Fo(R)q Fr(\(\()p Fp(m;)17 b Fr(0\))p Fp(;)g Fr(\()p Fp(\036;)g Fr(0\);)g Fp(z)t Fr(\))p Fp(\025)p Fr(\()p Fp(\036;)g(z)t Fr(\))2844 559 y Fw(x)2886 600 y Fp(:)0 1016 y Fr(Our)30 b(next)h(lemma)g(deals)h(with)e(a)g(random)h (p)s(oten)m(tial)f Fp(V)22 b Fr(.)43 b(Let)30 b(\(\012)p Fp(;)17 b Fd(B)p Fr(\))30 b(b)s(e)h(as)f(in)h(Section)g(1.2)f(and)g Fp(P)0 1137 y Fr(b)s(e)j(an)f(arbitrary)h(probabilit)m(y)g(measure)h (on)f(\(\012)p Fp(;)17 b Fd(B)p Fr(\).)0 1345 y Fh(Lemma)39 b(3.4)49 b Fg(L)-5 b(et)44 b Fr(\()p Fp(a;)17 b(b)p Fr(\))44 b Fg(b)-5 b(e)44 b(an)f(interval)h(such)f(that)h(for)g(some)f Fr(2)p Fp(=)p Fr(3)h Fp(<)g(s)g(<)g Fr(1)p Fg(,)i Fp(\016)i(>)c Fr(0)g Fg(and)f(al)5 b(l)0 1466 y Fp(m;)17 b(n)28 b Fo(2)g Fn(Z)p Fg(,)830 1659 y Fr(sup)724 1745 y Fw(e)p Fl(2)p Fv(\()p Fw(a;b)p Fv(\))p Fw(;\017)p Fl(6)p Fv(=0)1101 1659 y Fn(E)p Fr(\()p Fo(jR)p Fr(\(\()p Fp(m;)17 b Fr(0\))p Fp(;)g Fr(\()p Fp(n;)g Fr(0\);)g Fp(e)k Fr(+)h(i)p Fp(\017)p Fr(\))p Fo(j)2220 1618 y Fw(s)2257 1659 y Fr(\))27 b Fp(<)h(C)7 b Fo(h)p Fp(n)22 b Fo(\000)h Fp(m)p Fo(i)2846 1618 y Fl(\000)p Fv(1)p Fl(\000)p Fw(\016)3029 1659 y Fp(:)0 1922 y Fg(Then)34 b(for)h(al)5 b(l)34 b Fp(x)28 b Fo(2)g Fn(Z)792 1937 y Fv(+)887 1922 y Fg(ther)-5 b(e)35 b(is)f(a)h(c)-5 b(onstant)34 b Fp(C)1777 1937 y Fw(x)1856 1922 y Fg(such)h(that)845 2115 y Fr(sup)739 2201 y Fw(e)p Fl(2)p Fv(\()p Fw(a;b)p Fv(\))p Fw(;\017)p Fl(6)p Fv(=0)1116 2115 y Fn(E)p Fr(\()p Fo(jR)p Fr(\(\()p Fp(m;)17 b Fr(0\))p Fp(;)g Fr(\()p Fp(n;)g(x)p Fr(\);)g Fp(e)22 b Fr(+)g(i)p Fp(\017)p Fr(\))p Fo(j)2242 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Fw(k)364 4888 y Fj(1)399 4879 y Fw(;)p Fl(\001\001\001)10 b Fw(;k)546 4891 y Fe(l)569 4879 y Fl(2)p Ff(Z)710 4681 y Fm(\000)756 4762 y Fo(h)p Fp(k)846 4777 y Fv(1)907 4762 y Fo(\000)23 b Fp(m)p Fo(ih)p Fp(k)1221 4777 y Fv(2)1282 4762 y Fo(\000)f Fp(k)1432 4777 y Fv(1)1472 4762 y Fo(i)17 b Fp(:)g(:)g(:)e Fo(h)p Fp(k)1748 4777 y Fw(l)1796 4762 y Fo(\000)22 b Fp(k)1946 4777 y Fw(l)q Fl(\000)p Fv(1)2062 4762 y Fo(ih)p Fp(n)g Fo(\000)h Fp(k)2371 4777 y Fw(l)2396 4762 y Fo(i)2435 4681 y Fm(\001)2481 4703 y Fl(\000)p Fv(1)p Fl(\000)p Fw(\016)2691 4762 y Fo(\024)29 b Fp(C)2874 4720 y Fw(l)2900 4762 y Fo(h)p Fp(n)22 b Fo(\000)g Fp(m)p Fo(i)3242 4720 y Fl(\000)p Fv(1)p Fl(\000)p Fw(\016)3425 4762 y Fp(:)0 5206 y Fh(Pro)s(of.)65 b Fr(An)33 b(elemen)m(tary)h(induction.)45 b Fb(2)p eop %%Page: 16 16 16 15 bop 3682 100 a Fr(16)0 407 y Fi(3.2)135 b(The)45 b(prop)t(ert)l(y)g Fa(P)0 641 y Fr(Throughout)33 b(this)g(section)h(w)m (e)f(assume)i(that)d(the)h(conditions)h(of)e(Theorem)i(1.4)e(hold.)0 811 y(Let)39 b(\()p Fp(a;)17 b(b)p Fr(\))38 b Fo(\032)h Fr(\()p Fo(\000)p Fp(c)704 826 y Fw(d)745 811 y Fp(;)17 b(c)831 826 y Fw(d)871 811 y Fr(\).)62 b(W)-8 b(e)39 b(sa)m(y)h(that)e(the)i(prop)s(ert)m(y)f Fo(P)47 b Fr(holds)40 b(on)e(\()p Fp(a;)17 b(b)p Fr(\))39 b(if)g(for)f(all)h Fp(R)g Fo(\025)g Fr(0)f(and)0 932 y Fp( )32 b Fo(2)c(H)q Fr(,)952 1214 y(lim)924 1276 y Fw(T)10 b Fl(!1)1178 1146 y Fr(1)p 1143 1191 120 4 v 1143 1282 a(2)p Fp(T)1289 1078 y Fm(Z)1388 1104 y Fw(T)1344 1303 y Fl(\000)p Fw(T)1471 1214 y Fn(E)1554 1133 y Fm(\000)1599 1214 y Fo(k)p Fh(1)1705 1229 y Fw(R)1763 1214 y Fr(e)1806 1172 y Fl(\000)p Fv(i)p Fw(tH)1974 1214 y Fh(1)2030 1229 y Fv(\()p Fw(a;b)p Fv(\))2176 1214 y Fr(\()p Fp(H)e Fr(\))p Fp( )t Fo(k)2458 1172 y Fv(2)2497 1133 y Fm(\001)2559 1214 y Fr(d)p Fp(t)28 b Fr(=)f(0)p Fp(:)0 1493 y Fr(T)-8 b(o)27 b(pro)m(v)m(e)i(Theorem)f(1.4)f (it)g(su\016ces)i(to)e(sho)m(w)h(that)f(the)h(prop)s(ert)m(y)g Fo(P)35 b Fr(holds)28 b(on)f(ev)m(ery)i(in)m(terv)-5 b(al)28 b(\()p 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b(Lemma)g(3.2)f(yield)i(that)894 5087 y Fo(R)p Fr(\()p Fp(n;)17 b(z)t Fr(\))29 b(=)e Fp(j)6 b Fr(\()p Fp(n;)17 b(z)t Fr(\))23 b Fo(\000)1732 4992 y Fm(X)1785 5204 y Fw(k)1893 5087 y Fp(j)6 b Fr(\()p Fp(n)22 b Fo(\000)h Fp(k)s(;)17 b(z)t Fr(\))p Fp(V)22 b Fr(\()p Fp(k)s Fr(\))p Fo(R)p Fr(\()p Fp(k)s(;)17 b(z)t Fr(\))p Fp(:)p eop %%Page: 22 22 22 21 bop 3682 100 a Fr(22)0 407 y(Th)m(us,)514 639 y Fh(E)p Fr(\()p Fo(jR)p Fr(\()p Fp(n;)17 b(z)t Fr(\))p Fo(j)993 598 y Fw(s)1030 639 y Fr(\))27 b Fo(\024)h(j)p Fp(j)6 b Fr(\()p Fp(n;)17 b(z)t Fr(\))p Fo(j)1529 598 y Fw(s)1588 639 y Fr(+)1686 544 y Fm(X)1739 756 y Fw(k)1847 639 y Fo(j)p Fp(j)6 b Fr(\()p Fp(n)22 b Fo(\000)g Fp(k)s(;)17 b(z)t Fr(\))p Fo(j)2351 598 y Fw(s)2388 639 y Fh(E)p Fr(\()p Fo(j)p Fp(V)k Fr(\()p Fp(k)s Fr(\))p Fo(j)2764 598 y Fw(s)2800 639 y Fo(jR)p Fr(\()p Fp(k)s(;)c(z)t Fr(\))p Fo(j)3163 598 y Fw(s)3200 639 y Fr(\))p Fp(:)265 b Fr(\(3.33\))0 943 y(The)37 b(decoupling)g(lemma)f(of)f(Aizenman-Molc) 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Fl(\000)p Fv(3)p Fw(s=)p Fv(2)2386 4502 y Fp(:)0 4787 y Fr(This)34 b(yields)g(Theorem)g(3.12)e(with)1145 5023 y(\003)1213 5038 y Fw(s)1278 5023 y Fr(=)1381 4942 y Fm(\000)1427 5023 y Fr(min\()p Fp(C)1704 4981 y Fl(\000)p Fv(1)1697 5047 y(1)1799 5023 y Fp(;)17 b Fr(\()p Fp(C)1951 5038 y Fv(2)1989 5023 y Fp(C)2059 5038 y Fv(3)2099 5023 y Fr(\))2137 4981 y Fl(\000)p Fv(1)2231 5023 y Fr(\))p Fp(=K)2401 5038 y Fw(s)2437 4942 y Fm(\001)2483 4964 y Fv(1)p Fw(=s)2607 5023 y Fp(:)p eop %%Page: 23 23 23 22 bop 3682 100 a Fr(23)0 407 y Fq(References)0 667 y Fr([A])220 b(Aizenman,)49 b(M.:)69 b(Lo)s(calization)45 b(at)f(w)m(eak)i(disorder:)70 b(some)45 b(elemen)m(tary)i(b)s(ounds.)f Fg(R)-5 b(ev.)347 788 y(Math.)35 b(Phys.)d Fh(6)p Fr(,)h(1163)f (\(1994\))0 991 y([AM])131 b(Aizenman,)52 b(M.,)f(Molc)m(hano)m(v,)h (S.:)72 b(Lo)s(calization)47 b(at)g(large)f(disorder)i(and)f(at)g (extreme)347 1112 y(energies:)e(an)32 b(elemen)m(tary)j(deriv)-5 b(ation.)33 b Fg(Commun.)h(Math.)h(Phys.)d Fh(157)p Fr(,)h(245)f (\(1993\))0 1315 y([A)m(G])146 b(Aizenman,)34 b(M.,)f(Graf,)f(G.-M.:)43 b(Lo)s(calization)32 b(b)s(ounds)i(for)e(an)g(electron)i(gas.)e Fg(J.)j(Phys.)g(A)347 1435 y Fh(31)p Fr(,)e(6783)e(\(1998\))0 1639 y([CS])169 b(Chahrour,)69 b(A.,)h(Sah)m(bani,)f(J.:)102 b(On)62 b(the)g(sp)s(ectral)g(and)g(scattering)h(theory)f(of)f(the)347 1759 y(Sc)m(hr\177)-49 b(odinger)34 b(op)s(erator)e(with)h(surface)g(p) s(oten)m(tial.)g Fg(R)-5 b(ev.)35 b(Math.)g(Phys.)p Fr(,)d Fh(12)p Fr(,)h(561)f(\(2000\))0 1963 y([CFKS])49 b(Cycon,)c(H.,)g(F)-8 b(ro)s(ese,)44 b(R.,)g(Kirsc)m(h,)h(W.,)g(Simon,)f(B.:)62 b Fg(Schr\177)-50 b(odinger)42 b(Op)-5 b(er)g(ators.)41 b Fr(Berlin-)347 2083 y(Heidelb)s(erg:)j(Springer-V)-8 b(erlag)33 b(1987)0 2286 y([Gr])178 b(Graf,)38 b(G.-M.:)54 b(Anderson)40 b(lo)s(calization)e(and)g(the)g(space-time)i(c)m (haracteristic)g(of)d(con)m(tin-)347 2407 y(uum)c(states.)h Fg(J.)h(Stat.)g(Phys.)d Fh(75)p Fr(,)h(337)f(\(1994\))0 2610 y([Da])169 b(Da)m(vies,)29 b(E.B.:)41 b 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b(ain)m(b)s(erg,)33 b(B.:)44 b(unpublished)0 1905 y([P])227 b(P)m(astur,)26 b(L.:)40 b(Surface)24 b(w)m(a)m(v)m(es:)42 b(propagation)23 b(and)h(lo)s(calization.)g Fg(Journ)n(\023)-47 b(ees)25 b(\\)3200 1880 y(\023)3186 1905 y(Equations)i(aux)347 2026 y(d)n(\023)-47 b(eriv)n(\023)g(ees)39 b(p)-5 b(artiel)5 b(les")41 b(\(Saint-Je)-5 b(an-de)40 b(Monts,)k(1995\))p Fr(,)d(Exp.)g(No.)g(VI,)3125 2001 y(\023)3116 2026 y(Ecole)g(P)m(olytec)m(h.)347 2146 y(P)m(alaisseau)34 b(\(1995\))0 2350 y([RS])167 b(Reed,)33 b(M.,)h(Simon,)f(B.:)44 b Fg(Metho)-5 b(ds)35 b(of)g(Mo)-5 b(dern)35 b(Mathematic)-5 b(al)35 b(Physics,)f(IV.)h(A)n(nalysis)g(of)347 2470 y(Op)-5 b(er)g(ators)p Fr(,)32 b(London:)43 b(Academic)35 b(Press)f(\(1978\))0 2673 y([S])239 b(Simon,)42 b(B.:)58 b(Sp)s(ectral)40 b(analysis)h(of)e(rank)h(one)g(p)s(erturbations)g(and) g(applications.)g Fg(CRM)347 2794 y(Pr)-5 b(o)g(c.)34 b(L)-5 b(e)g(ctur)g(e)36 b(Notes)p Fr(,)d(Pro)m(vidence,)i(RI:)d Fh(8)p Fr(,)h(1995)p eop %%Trailer end userdict 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